RELATIVITY 


ALBERT  EINSTEIN 


^OFPR?S^ 
(  MAY  2  1933  J 
X^OeiCALSt)^^ 


Division 


Section 


Digitized  by  the  Internet  Archive 
in  2019  with  funding  from 
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https://archive.org/details/rneaningofrelativ00eins_0 


THE  MEANING  OF  RELATIVITY 


THE  MEANING  OF 


RELATIVITY^' 


OF  PRlNi 


FOUR  LECTURES  DELIVERED  A 
PRINCETON  UNIVERSITY,  MAY,  1921 


A 


VUY  9  10' 

‘  r\  I  AJ 


//£ 

<^0GIGAL8^ 


BY  / 

ALBERT  EINSTEIN 


WITH  FOUR  DIAGRAMS 


PRINCETON 

PRINCETON  UNIVERSITY  PRESS 

1923 


Copyright  1922 
Princeton  University  Press 
Published  iq22 


PRINTED  IN  GREAT  BRITAIN 
AT  THE  ABERDEEN  UNIVERSITY  PRESS 
ABERDEEN 


Note. — The  translation  of  these  lectures  into  English 
was  made  by  Edwin  Plimpton  Adams,  Professor 
of  Physics  in  Princeton  University 


CONTENTS 


LECTURE  I 

Space  and  Time  in  Pre-Relativity  Physics 


LECTURE  II 

The  Theory  of  Special  Relativity 


LECTURE  III 

The  General  Theory  of  Relativity 


LECTURE  IV 

The  General  Theory  of  Relativity  ( co?itinued ) 


Index . 


THE  MEANING  OF  RELATIVITY 


LECTURE  I 


SPACE  AND  TIME  IN  PRE-RELATIVITY 

PHYSICS 


HE  theory  of  relativity  is  intimately  connected  with 


-1  the  theory  of  space  and  time.  I  shall  therefore  begin 
with  a  brief  investigation  of  the  origin  of  our  ideas  of  space 
and  time,  although  in  doing  so  I  know  that  I  introduce  a 
controversial  subject.  The  object  of  all  science,  whether 
natural  science  or  psychology,  is  to  co-ordinate  our  experi¬ 
ences  and  to  bring  them  into  a  logical  system.  How  are 
our  customary  ideas  of  space  and  time  related  to  the 
character  of  our  experiences  ? 

The  experiences  of  an  individual  appear  to  us  arranged 
in  a  series  of  events  ;  in  this  series  the  single  events  which 
we  remember  appear  to  be  ordered  according  to  the  criterion 
of  “  earlier  ”  and  “  later,”  which  cannot  be  analysed  further. 
There  exists,  therefore,  for  the  individual,  an  I -time,  or 
subjective  time.  This  in  itself  is  not  measurable.  I  can, 
indeed,  associate  numbers  with  the  events,  in  such  a  way 
that  a  greater  number  is  associated  with  the  later  event 
than  with  an  earlier  one  ;  but  the  nature  of  this  association 
may  be  quite  arbitrary.  This  association  I  can  define  by 
means  of  a  clock  by  comparing  the  order  of  events  furnished 


i 


2  THE  MEANING  OF  RELATIVITY 


by  the  clock  with  the  order  of  the  given  series  of  events. 
We  understand  by  a  clock  something  which  provides  a 
series  of  events  which  can  be  counted,  and  which  has  other 
properties  of  which  we  shall  speak  later. 

By  the  aid  of  speech  different  individuals  can,  to  a  certain 
extent,  compare  their  experiences.  In  this  way  it  is  shown 
that  certain  sense  perceptions  of  different  individuals 
correspond  to  each  other,  while  for  other  sense  perceptions 
no  such  correspondence  can  be  established.  We  are  ac¬ 
customed  to  regard  as  real  those  sense  perceptions  which 
are  common  to  different  individuals,  and  which  therefore 
are,  in  a  measure,  impersonal.  The  natural  sciences,  and 
in  particular,  the  most  fundamental  of  them,  physics,  deal 
with  such  sense  perceptions.  The  conception  of  physical 
bodies,  in  particular  of  rigid  bodies,  is  a  relatively  constant 
complex  of  such  sense  perceptions.  A  clock  is  also  a  body, 
or  a  system,  in  the  same  sense,  with  the  additional  property 
that  the  series  of  events  which  it  counts  is  formed  of 
elements  all  of  which  can  be  regarded  as  equal. 

The  only  justification  for  our  concepts  and  system  of 
concepts  is  that  they  serve  to  represent  the  complex  of 
our  experiences;  beyond  this  they  have  no  legitimacy.  I 
am  convinced  that  the  philosophers  have  had  a  harmful 
effect  upon  the  progress  of  scientific  thinking  in  removing 
certain  fundamental  concepts  from  the  domain  of  empiric¬ 
ism,  where  they  are  under  our  control,  to  the  intangible 
heights  of  the  a  priori.  For  even  if  it  should  appear  that 
the  universe  of  ideas  cannot  be  deduced  from  experience 
by  logical  means,  but  is,  in  a  sense,  a  creation  of  the  human 
mind,  without  which  no  science  is  possible,  nevertheless 


PRE-RELATIVITY  PHYSICS 


3 


this  universe  of  ideas  is  just  as  little  independent  of  the 
nature  of  our  experiences  as  clothes  are  of  the  form  of 
the  human  body.  This  is  particularly  true  of  our  con¬ 
cepts  of  time  and  space,  which  physicists  have  been 
obliged  by  the  facts  to  bring  down  from  the  Olympus  of 
the  a  priori  in  order  to  adjust  them  and  put  them  in  a 
serviceable  condition. 

We  now  come  to  our  concepts  and  judgments  of  space. 
It  is  essential  here  also  to  pay  strict  attention  to  the 
relation  of  experience  to  our  concepts.  It  seems  to  me 
that  Poincare  clearly  recognized  the  truth  in  the  account 
he  gave  in  his  book,  “  La  Science  et  l’Hypothese.” 
Among  all  the  changes  which  we  can  perceive  in  a  rigid 
body  those  are  marked  by  their  simplicity  which  can  be 
made  reversibly  by  an  arbitrary  motion  of  the  body; 
Poincare  calls  these,  changes  in  position.  By  means  of 
simple  changes  in  position  we  can  bring  two  bodies  into 
contact.  The  theorems  of  congruence,  fundamental  in 
geometry,  have  to  do  with  the  laws  that  govern  such 
changes  in  position.  For  the  concept  of  space  the  follow¬ 
ing  seems  essential.  We  can  form  new  bodies  by  bringing 
bodies  B,  C,  ...  up  to  body  A  ;  we  say  that  we  continue 
body  A.  We  can  continue  body  A  in  such  a  way  that 
it  comes  into  contact  with  any  other  body,  X.  The 
ensemble  of  all  continuations  of  body  A  we  can  designate 
as  the  “space  of  the  body  A.”  Then  it  is  true  that  all 
bodies  are  in  the  “space  of  the  (arbitrarily  chosen)  body 
AA  In  this  sense  we  cannot  speak  of  space  in  the 
abstract,  but  only  of  the  “space  belonging  to  a  body  AA 
The  earth’s  crust  plays  such  a  dominant  role  in  our  daily 


4  THE  MEANING  OF  RELATIVITY 


life  in  judging  the  relative  positions  of  bodies  that  it  has 
led  to  an  abstract  conception  of  space  which  certainly 
cannot  be  defended.  In  order  to  free  ourselves  from  this 
fatal  error  we  shall  speak  only  of  “bodies  of  reference,” 
or  “  space  of  reference.”  It  was  only  through  the  theory 
of  general  relativity  that  refinement  of  these  concepts 
became  necessary,  as  we  shall  see  later. 

I  shall  not  go  into  detail  concerning  those  properties 
of  the  space  of  reference  which  lead  to  our  conceiving 
points  as  elements  of  space,  and  space  as  a  continuum. 
Nor  shall  I  attempt  to  analyse  further  the  properties  of 
space  which  justify  the  conception  of  continuous  series 
of  points,  or  lines.  If  these  concepts  are  assumed,  together 
with  their  relation  to  the  solid  bodies  of  experience,  then 
it  is  easy  to  say  what  we  mean  by  the  three-dimensionality 
of  space ;  to  each  point  three  numbers,  xv  x2,  x3  (co¬ 
ordinates),  may  be  associated,  in  such  a  way  that  this 
association  is  uniquely  reciprocal,  and  that  xv  xv  and  x2 
vary  continuously  when  the  point  describes  a  continuous 
series  of  points  (a  line). 

It  is  assumed  in  pre-relativity  physics  that  the  laws  of 
the  orientation  of  ideal  rigid  bodies  are  consistent  with 
Euclidean  geometry.  What  this  means  may  be  expressed 
as  follows :  Two  points  marked  on  a  rigid  body  form 
an  interval.  Such  an  interval  can  be  oriented  at  rest, 
relatively  to  our  space  of  reference,  in  a  multiplicity  of 
ways.  If,  now,  the  points  of  this  space  can  be  referred 
to  co-ordinates^,  xv  xv  in  such  a  way  that  the  differences 
of  the  co-ordinates,  Axv  Aq,  Ar3,  of  the  two  ends  of  the 
interval  furnish  the  same  sum  of  squares, 

s 2  =  A*i2  +  Ax22  +  kx3  .  .  (i) 


PRE-RELATIVITY  PHYSICS 


5 


for  every  orientation  of  the  interval,  then  the  space  of 
reference  is  called  Euclidean,  and  the  co-ordinates 
Cartesian.*  It  is  sufficient,  indeed,  to  make  this  assump¬ 
tion  in  the  limit  for  an  infinitely  small  interval.  Involved 
in  this  assumption  there  are  some  which  are  rather  less 
special,  to  which  we  must  call  attention  on  account  of 
their  fundamental  significance.  In  the  first  place,  it  is 
assumed  that  one  can  move  an  ideal  rigid  body  in  an 
arbitrary  manner.  In  the  second  place,  it  is  assumed 
that  the  behaviour  of  ideal  rigid  bodies  towards  orienta¬ 
tion  is  independent  of  the  material  of  the  bodies  and  their 
changes  of  position,  in  the  sense  that  if  two  intervals  can 
once  be  brought  into  coincidence,  they  can  always  and 
everywhere  be  brought  into  coincidence.  Both  of  these 
assumptions,  which  are  of  fundamental  importance  for 
geometry  and  especially  for  physical  measurements, 
naturally  arise  from  experience  ;  in  the  theory  of  general 
relativity  their  validity  needs  to  be  assumed  only  for 
bodies  and  spaces  of  reference  which  are  infinitely  small 
compared  to  astronomical  dimensions. 

The  quantity  s  we  call  the  length  of  the  interval.  In 
order  that  this  may  be  uniquely  determined  it  is  necessary 
to  fix  arbitrarily  the  length  of  a  definite  interval ;  for 
example,  we  can  put  it  equal  to  I  (unit  of  length).  Then 
the  lengths  of  all  other  intervals  may  be  determined.  If 
we  make  the  xv  linearly  dependent  upon  a  parameter  X, 

xv  =  dv  +  X^„, 

*  This  relation  must  hold  for  an  arbitrary  choice  of  the  origin  and  of  the 
direction  (ratios  Axl  :  Ax2  :  Ax3)  of  the  interval. 


6  THE  MEANING  OF  RELATIVITY 


we  obtain  a  line  which  has  all  the  properties  of  the  straight 
lines  of  the  Euclidean  geometry.  In  particular,  it  easily 
follows  that  by  laying  off  n  times  the  interval  n  upon  a 
straight  line,  an  interval  of  length  n's  is  obtained.  A 
length,  therefore,  means  the  result  of  a  measurement 
carried  out  along  a  straight  line  by  means  of  a  unit 
measuring  rod.  It  has  a  significance  which  is  as  inde¬ 
pendent  of  the  system  of  co-ordinates  as  that  of  a  straight 
line,  as  will  appear  in  the  sequel. 

We  come  now  to  a  train  of  thought  which  plays  an 
analogous  role  in  the  theories  of  special  and  general 
relativity.  We  ask  the  question  :  besides  the  Cartesian 
co-ordinates  which  we  have  used  are  there  other  equivalent 
co-ordinates  ?  An  interval  has  a  physical  meaning  which 
is  independent  of  the  choice  of  co-ordinates ;  and  so  has 
the  spherical  surface  which  we  obtain  as  the  locus  of  the 
end  points  of  all  equal  intervals  that  we  lay  off  from  an 
arbitrary  point  of  our  space  of  reference.  If  xv  as  well  as 
x  v  {y  from  I  to  3)  are  Cartesian  co-ordinates  of  our  space 
of  reference,  then  the  spherical  surface  will  be  expressed 
in  our  two  systems  of  co-ordinates  by  the  equations 

A^2  =  const.  .  .  (2) 

=  const.  .  .  .  (2a) 

How  must  the  x v  be  expressed  in  terms  of  thexv  in  order 
that  equations  (2)  and  (2a)  may  be  equivalent  to  each 
other  ?  Regarding  the  x  v  expressed  as  functions  of  the 
xvy  we  can  write,  by  Taylor’s  theorem,  for  small  values  of 
the  Axu, 


PRE-RELATIVITY  PHYSICS 


7 


AY.  =  J 


dx. 


Ax„ 


i . 

+  5. 


a/3 


c)Vv 


Axai\xp  . 


If  we  substitute  (2a)  in  this  equation  and  compare  with 
(i),  we  see  that  the  x  v  must  be  linear  functions  of  the  xv. 
If  we  therefore  put 

x  v  —  av  +  bvaxa  .  .  .  (3) 

a 

or  Ar'„  =  ^<Ara  .  •  •  •  (3a) 


then  the  equivalence  of  equations  (2)  and  (2a)  is  expressed 
in  the  form 


=  \^Jix2  (X  independent  of  Axv)  .  (2b) 

It  therefore  follows  that  X  must  be  a  constant.  If  we  put 
X  =  1,  (2b)  and  (3a)  furnish  the  conditions 


j3  ^aj8  •  •  •  (4) 


in  which  Sai3  =  1,  cr  8af}  =  o,  according  as  a  =  /3  or 
a  /3.  The  conditions  (4)  are  called  the  conditions  of  ortho¬ 
gonality,  and  the  transformations  (3),  (4),  linear  orthogonal 

transformations.  If  we  stipulate  that  s'1  =  ^Ax2  shall  be 

equal  to  the  square  of  the  length  in  every  system  of 
co-ordinates,  and  if  we  always  measure  with  the  same  unit 
scale,  then  X  must  be  equal  to  1.  Therefore  the  linear 
orthogonal  transformations  are  the  only  ones  by  means  of 
which  we  can  pass  from  one  Cartesian  system  of  co¬ 
ordinates  in  our  space  of  reference  to  another.  We  see 


8  THE  MEANING  OF  RELATIVITY 


that  in  applying  such  transformations  the  equations  of 
a  straight  line  become  equations  of  a  straight  line. 
Reversing  equations  (3a)  by  multiplying  both  sides  by  bvfi 
and  summing  for  all  the  vs,  we  obtain 

v  =  ^ b vofo a.  ~  /  $ap/\Xa  =  l\X p  .  (5) 

va  a 

The  same  coefficients,  b,  also  determine  the  inverse 
substitution  of  Axv.  Geometrically,  bva  is  the  cosine  of  the 
angle  between  the  x  v  axis  and  the  ;ra  axis. 

To  sum  up,  we  can  say  that  in  the  Euclidean  geometry 
there  are  (in  a  given  space  of  reference)  preferred  systems 
of  co-ordinates,  the  Cartesian  systems,  which  transform 
into  each  other  by  linear  orthogonal  transformations. 
The  distance  s  between  two  points  of  our  space  of 
reference,  measured  by  a  measuring  rod,  is  expressed  in 
such  co-ordinates  in  a  particularly  simple  manner.  The 
whole  of  geometry  may  be  founded  upon  this  conception 
of  distance.  In  the  present  treatment,  geometry  is 
related  to  actual  things  (rigid  bodies),  and  its  theorems 
are  statements  concerning  the  behaviour  of  these  things, 
which  may  prove  to  be  true  or  false. 

One  is  ordinarily  accustomed  to  study  geometry 
divorced  from  any  relation  between  its  concepts  and 
experience.  There  are  advantages  in  isolating  that 
which  is  purely  logical  and  independent  of  what  is,  in 
principle,  incomplete  empiricism.  This  is  satisfactory  to 
the  pure  mathematician.  He  is  satisfied  if  he  can  deduce 
his  theorems  from  axioms  correctly,  that  is,  without 
errors  of  logic.  The  question  as  to  whether  Euclidean 


PRE-RELATIVITY  PHYSICS 


9 


geometry  is  true  or  not  does  not  concern  him.  But  for 
our  purpose  it  is  necessary  to  associate  the  fundamental 
concepts  of  geometry  with  natural  objects  ;  without  such 
an  association  geometry  is  worthless  for  the  physicist. 
The  physicist  is  concerned  with  the  question  as  to 
whether  the  theorems  of  geometry  are  true  or  not.  That 
Euclidean  geometry,  from  this  point  of  view,  affirms 
something  more  than  the  mere  deductions  derived 
logically  from  definitions  may  be  seen  from  the  following 
simple  consideration. 


Between  n  points  of  space  there  are 


distances, 


;  between  these  and  the  3 n  co-ordinates  we  have  the 
relations 


S^v“  —  X\ (t-))2  *1"  C*-2(M)  '*2(»'))  +  •  •  • 

n(n  -  1)  . 

From  these  -  equations  the  3^  co-ordinates 

may  be  eliminated,  and  from  this  elimination  at  least 
n(n  -  1) 

- -  3 n  equations  in  the  s will  result.*  Since 

the  are  measurable  quantities,  and  by  definition  are 
independent  of  each  other,  these  relations  between  the 
s^v  are  not  necessary  a  priori. 

From  the  foregoing  it  is  evident  that  the  equations  of 
transformation  (3),  (4)  have  a  fundamental  significance  in 
Euclidean  geometry,  in  that  they  govern  the  transforma¬ 
tion  from  one  Cartesian  system  of  co-ordinates  to  another. 
The  Cartesian  systems  of  co-ordinates  are  characterized 

n(n  -  1) 


In  reality  there  are 


-  3«  +  6  equations. 


10  THE  MEANING  OF  RELATIVITY 


by  the  property  that  in  them  the  measurable  distance 
between  two  points,  s ,  is  expressed  by  the  equation 

If  K(Xv)  and  K\Xv)  are  two  Cartesian  systems  of  co¬ 
ordinates,  then 

^>Auy2  =  ^Aa'V2. 

The  right-hand  side  is  identically  equal  to  the  left-hand 


side  on  account  of  the  equations  of  the  linear  orthogonal 
transformation,  and  the  right-hand  side  differs  from  the 
left-hand  side  only  in  that  the  xv  are  replaced  by  the  x  v. 

This  is  expressed  by  the  statement  that  is  an 

invariant  with  respect  to  linear  orthogonal  transforma¬ 
tions.  It  is  evident  that  in  the  Euclidean  geometry  only 
such,  and  all  such,  quantities  have  an  objective  signifi¬ 
cance,  independent  of  the  particular  choice  of  the  Cartesian 
co-ordinates,  as  can  be  expressed  by  an  invariant  with 
respect  to  linear  orthogonal  transformations.  This  is  the 
reason  that  the  theory  of  invariants,  which  has  to  do  with 
the  laws  that  govern  the  form  of  invariants,  is  so  important 
for  analytical  geometry. 

As  a  second  example  of  a  geometrical  invariant,  con¬ 
sider  a  volume.  This  is  expressed  by 


V  —  j  j  jdz1dx2dx3. 

By  means  of  Jacobi’s  theorem  we  may  write 


dx \dx \dx 3 


ill 


ap'i,  x'.2,  x3) 
x.2,  x3) 


dxYdx2dx3 


PRE-RELATIVITY  PHYSICS 


11 


where  the  integrand  in  the  last  integral  is  the  functional 
determinant  of  the  x  v  with  respect  to  the  xv,  and  this  by 
(3)  is  equal  to  the  determinant  |  b^v  |  of  the  coefficients 
of  substitution,  bV0L.  If  we  form  the  determinant  of  the 
S,Aa  from  equation  (4),  we  obtain,  by  means  of  the  theorem 
of  multiplication  of  determinants, 


v 


If  we  limit  ourselves  to  those  transformations  which  have 
the  determinant  +  I,*  and  only  these  arise  from  con¬ 
tinuous  variations  of  the  systems  of  co-ordinates,  then  V 
is  an  invariant. 

Invariants,  however,  are  not  the  only  forms  by  means 
of  which  we  can  give  expression  to  the  independence  of 
the  particular  choice  of  the  Cartesian  co-ordinates.  Vectors 
and  tensors  are  other  forms  of  expression.  Let  us  express 
the  fact  that  the  point  with  the  current  co-ordinates  xv  lies 
upon  a  straight  line.  We  have 


xv  -  Av  =  \BV  (v  from  1  to  3). 

Without  limiting  the  generality  we  can  put 

]>A.2  =  1. 

If  we  multiply  the  equations  by  b^v  (compare  (3a)  and 
(5))  and  sum  for  all  the  p’s,  we  get 

x  p  —  A  p  =  \B  p 

*  There  are  thus  two  kinds  of  Cartesian  systems  which  are  designated 
as  “right-handed”  and  “left-handed”  systems.  The  difference  between 
these  is  familiar  to  every  physicist  and  engineer.  It  is  interesting  to  note 
that  these  two  kinds  of  systems  cannot  be  defined  geometrically,  but  only 
the  contrast  between  them. 


12  THE  MEANING  OF  RELATIVITY 


where  we  have  written 

^0  =  y  bpvBv ;  Ap  =  '^bpvAv. 

V  V 

These  are  the  equations  of  straight  lines  with  respect 
to  a  second  Cartesian  system  of  co-ordinates  K'.  They 
have  the  same  form  as  the  equations  with  respect  to  the 
original  system  of  co-ordinates.  It  is  therefore  evident 
that  straight  lines  have  a  significance  which  is  independent 
of  the  system  of  co-ordinates.  Formally,  this  depends 
upon  the  fact  that  the  quantities  (xv  -  A v)  -  \BV  are 
transformed  as  the  components  of  an  interval,  The 

ensemble  of  three  quantities,  defined  for  every  system  of 
Cartesian  co-ordinates,  and  which  transform  as  the  com¬ 
ponents  of  an  interval,  is  called  a  vector.  If  the  three 
components  of  a  vector  vanish  for  one  system  of  Cartesian 
co-ordinates,  they  vanish  for  all  systems,  because  the  equa¬ 
tions  of  transformation  are  homogeneous.  We  can  thus 
get  the  meaning  of  the  concept  of  a  vector  without  referring 
to  a  geometrical  representation.  This  behaviour  of  the 
equations  of  a  straight  line  can  be  expressed  by  saying 
that  the  equation  of  a  straight  line  is  co-variant  with  respect 
to  linear  orthogonal  transformations. 

We  shall  now  show  briefly  that  there  are  geometrical 
entities  which  lead  to  the  concept  of  tensors.  Let  P0  be 
the  centre  of  a  surface  of  the  second  degree,  P  any  point 
on  the  surface,  and  the  projections  of  the  interval  P0P 
upon  the  co-ordinate  axes.  Then  the  equation  of  the 
surface  is 


PRE-RELATIVITY  PHYSICS 


13 


In  this,  and  in  analogous  cases,  we  shall  omit  the  sign  of 
summation,  and  understand  that  the  summation  is  to  be 
carried  out  for  those  indices  that  appear  twice.  We  thus 
write  the  equation  of  the  surface 

The  quantities  a^v  determine  the  surface  completely,  for 
a  given  position  of  the  centre,  with  respect  to  the  chosen 
system  of  Cartesian  co-ordinates.  From  the  known  law 
of  transformation  for  the  (3a)  for  linear  orthogonal 
transformations,  we  easily  find  the  law  of  transformation 
for  the  a^v  *  : 

^  err  ' 

This  transformation  is  homogeneous  and  of  the  first  degree 
in  the  a^v.  On  account  of  this  transformation,  the  a^v 
are  called  components  of  a  tensor  of  the  second  rank  (the 
latter  on  account  of  the  double  index).  If  all  the  com¬ 
ponents,  of  a  tensor  with  respect  to  any  system  of 
Cartesian  co-ordinates  vanish,  they  vanish  with  respect  to 
every  other  Cartesian  system.  The  form  and  the  position 
of  the  surface  of  the  second  degree  is  described  by  this 
tensor  (a). 

Analytic  tensors  of  higher  rank  (number  of  indices) 
may  be  defined.  It  is  possible  and  advantageous  to 
regard  vectors  as  tensors  of  rank  1,  and  invariants  (scalars) 
as  tensors  of  rank  o.  In  this  respect,  the  problem  of  the 
theory  of  invariants  may  be  so  formulated  :  according  to 
what  laws  may  new  tensors  be  formed  from  given  tensors  ? 

*  The  equation  aVrlV^'r  =  1  may,  by  (5),  be  replaced  by  &’  errb  fxa-bpT^o-^T 
=  i,  from  which  the  result  stated  immediately  follows. 


14  THE  MEANING  OF  RELATIVITY 


We  shall  consider  these  laws  now,  in  order  to  be  able  to 
apply  them  later.  We  shall  deal  first  only  with  the 
properties  of  tensors  with  respect  to  the  transformation 
from  one  Cartesian  system  to  another  in  the  same  space 
of  reference,  by  means  of  linear  orthogonal  transforma¬ 
tions.  As  the  laws  are  wholly  independent  of  the  number 
of  dimensions,  we  shall  leave  this  number,  n,  indefinite  at 
first. 

Definition.  If  a  figure  is  defined  with  respect  to  every 
system  of  Cartesian  co-ordinates  in  a  space  of  reference  of 
n  dimensions  by  the  n a  numbers  A^p  .  .  .  (a  =  number 
of  indices),  then  these  numbers  are  the  components  of  a 
tensor  of  rank  a  if  the  transformation  law  is 


i u.'v'p'  •  *  •  ^ \ u.'p.^v'v^p’p  ■  *  •  ^ju .vp  *  •  *  •  (7 ) 

Remark.  From  this  definition  it  follows  that 


jj.vp  •  •  *  ^  fid yD  p  ...  .  .  (^) 

is  an  invariant,  provided  that  ( B ),  (Q,  (Z?)  .  .  .  are 
vectors.  Conversely,  the  tensor  character  of  ( A )  may  be 
inferred,  if  it  is  known  that  the  expression  (8)  leads  to  an 
invariant  for  an  arbitrary  choice  of  the  vectors  ( B ),  (C), 
etc. 

Addition  and  Subtraction.  By  addition  and  subtraction 
of  the  corresponding  components  of  tensors  of  the  same 
rank,  a  tensor  of  equal  rank  results  : 


A 


±  B 


—  P-Vp 


The  proof  follows  from  the  definition  of  a  tensor  given 
above. 

Multiplication.  From  a  tensor  of  rank  a  and  a  tensor 


PRE-RELATIVITY  PHYSICS 


15 


of  rank  /3  we  may  obtain  a  tensor  of  rank  a  +  (3  by 
multiplying  all  the  components  of  the  first  tensor  by  all 
the  components  of  the  second  tensor  : 

•  •  •  afi  •  •  •  •  •  •  -^a/3y  •  •  •  ( ^  O) 


Contraction.  A  tensor  of  rank  a  -  2  may  be  obtained 
from  one  of  rank  a  by  putting  two  definite  indices  equal 
to  each  other  and  then  summing  for  this  single  index  : 


T  —  a  ( - 

p  •  •  •  •  •  •  v 


Y A 

y  ■'■Vmp 


•  •  •)  •  (ii) 


The  proof  is 


A'  —  h  h  h  A 

-*1  fxp.p  •  *  •  t//uat/p.j3c/py  •  •  •  •‘-1  afiy  ’  • 


^afi^py  •  ••  A 

=  ...  A 


afiy 

a  ay 


In  addition  to  these  elementary  rules  of  operation 
there  is  also  the  formation  of  tensors  by  differentiation 
(“  erweiterung  ”) : 

(12) 


T, 


■p.vp 


'P 


New  tensors,  in  respect  to  linear  orthogonal  transforma¬ 
tions,  may  be  formed  from  tensors  according  to  these  rules 
of  operation. 

Symmetrical  Properties  of  Tensors.  Tensors  are  called 
symmetrical  or  skew-symmetrical  in  respect  to  two  of 
their  indices,  ^  and  v ,  if  both  the  components  which  result 
from  interchanging  the  indices  and  v  are  equal  to  each 
other  or  equal  with  opposite  signs. 

Condition  for  symmetry :  Apvp  =  Apvp. 

Condition  for  skew-symmetry:  Ap„p  =  -  A,,pp. 

Theorem.  The  character  of  symmetry  or  skew-symmetry 
exists  independently  of  the  choice  of  co-ordinates,  and  in 


16  THE  MEANING  OF  RELATIVITY 


this  lies  its  importance.  The  proof  follows  from  the 
equation  defining  tensors. 

Special  Tensors. 

I.  The  quantities  8pcr  (4)  are  tensor  components  (funda¬ 
mental  tensor). 

Proof.  If  in  the  right-hand  side  of  the  equation  of 
transformation  A\v  =  b^abvfiAa^  we  substitute  for  Aafi  the 
quantities  8afi  (which  are  equal  to  I  or  o  according  as 
a  =  ft  or  a  /3),  we  get 

/]'  _  h  h  _  £ 

The  justification  for  the  last  sign  of  equality  becomes 
evident  if  one  applies  (4)  to  the  inverse  substitution  (5). 

II.  There  is  a  tensor  (8^vp  .  .  .)  skew-symmetrical  with 
respect  to  all  pairs  of  indices,  whose  rank  is  equal  to  the 
number  of  dimensions,  n,  and  whose  components  are 
equal  to  +  I  or  -  1  according  as  [xvp  ...  is  an  even 
or  odd  permutation  of  123  .  .  . 

The  proof  follows  with  the  aid  of  the  theorem  proved 
above  \  bpa\  =  1. 

These  few  simple  theorems  form  the  apparatus  from 
the  theory  of  invariants  for  building  the  equations  of  pre¬ 
relativity  physics  and  the  theory  of  special  relativity. 

We  have  seen  that  in  pre-relativity  physics,  in  order  to 
specify  relations  in  space,  a  body  of  reference,  or  a  space 
of  reference,  is  required,  and,  in  addition,  a  Cartesian 
system  of  co-ordinates.  We  can  fuse  both  these  concepts 
into  a  single  one  by  thinking  of  a  Cartesian  system  of 
co-ordinates  as  a  cubical  frame- work  formed  of  rods  each 
of  unit  length.  The  co-ordinates  of  the  lattice  points  of 


PRE-RELATIVITY  PHYSICS 


17 


this  frame  are  integral  numbers.  It  follows  from  the 
fundamental  relation 

s2  =  Arp  +  Ar22  +  Ar32 

that  the  members  of  such  a  space-lattice  are  all  of  unit 
length.  To  specify  relations  in  time,  we  require  in 
addition  a  standard  clock  placed  at  the  origin  of  our 
Cartesian  system  of  co-ordinates  or  frame  of  reference. 
If  an  event  takes  place  anywhere  we  can  assign  to  it  three 
co-ordinates,  xvi  and  a  time  t,  as  soon  as  we  have  specified 
the  time  of  the  clock  at  the  origin  which  is  simultaneous 
with  the  event.  We  therefore  give  an  objective  signifi¬ 
cance  to  the  statement  of  the  simultaneity  of  distant 
events,  while  previously  we  have  been  concerned  only 
with  the  simultaneity  of  two  experiences  of  an  individual. 
The  time  so  specified  is  at  all  events  independent  of  the 
position  of  the  system  of  co-ordinates  in  our  space  of 
reference,  and  is  therefore  an  invariant  with  respect  to 
the  transformation  (3). 

It  is  postulated  that  the  system  of  equations  expressing 
the  laws  of  pre-relativity  physics  is  co-variant  with  respect 
to  the  transformation  (3),  as  are  the  relations  of  Euclidean 
geometry.  The  isotropy  and  homogeneity  of  space  is 
expressed  in  this  way.*  We  shall  now  consider  some  of 

*  The  laws  of  physics  could  be  expressed,  even  in  case  there  were  a 
unique  direction  in  space,  in  such  a  way  as  to  be  co-variant  with  respect  to 
the  transformation  (3) ;  but  such  an  expression  would  in  this  case  be  un¬ 
suitable.  If  there  were  a  unique  direction  in  space  it  would  simplify  the 
description  of  natural  phenomena  to  orient  the  system  of  co-ordinates  in  a 
definite  way  in  this  direction.  But  if,  on  the  other  hand,  there  is  no  unique 
direction  in  space  it  is  not  logical  to  formulate  the  laws  of  nature  in  such 
a  way  as  to  conceal  the  equivalence  of  systems  of  co-ordinates  that  are 


2 


18  THE  MEANING  OF  RELATIVITY 


the  more  important  equations  of  physics  from  this  point 
of  view. 

The  equations  of  motion  of  a  material  particle  are 


**.  y 

m~d¥.  “ 


(14) 


(dxv)  is  a  vector  ;  dt ,  and  therefore  also  an  invariant ; 

thus  (^r)  is  a  vector  ;  in  the  same  way  it  may  be  shown 

/  dsx  \ 

that  is  a  vector.  In  general,  the  operation  of  dif¬ 

ferentiation  with  respect  to  time  does  not  alter  the  tensor 
character.  Since  in  is  an  invariant  (tensor  of  rank  o), 

f  d2xv\ 

\l~df  )  lS  a  veci;or’  or  tensor  of  rank  I  (by  the  theorem 

of  the  multiplication  of  tensors).  If  the  force  (Av)  has 
a  vector  character,  the  same  holds  for  the  difference 

(d^x  \ 

m~d¥  ~  Xv'  ^ese  equations  of  motion  are  therefore 

valid  in  every  other  system  of  Cartesian  co-ordinates  in 
the  space  of  reference.  In  the  case  where  the  forces  are 
conservative  we  can  easily  recognize  the  vector  character 
of  (Xv).  For  a  potential  energy,  <F,  exists,  which  depends 
only  upon  the  mutual  distances  of  the  particles,  and  is 
therefore  an  invariant.  The  vector  character  of  the  force, 

Xv  =  -  ^7,  is  then  a  consequence  of  our  general  theorem 
about  the  derivative  of  a  tensor  of  rank  o. 


oriented  differently.  We  shall  meet  with  this  point  of  view  again  in  the 
theories  of  special  and  general  relativity. 


PRE-RELATIVITY  PHYSICS 


19 


Multiplying  by  the  velocity,  a  tensor  of  rank  i,  we 
obtain  the  tensor  equation 


dlx 


m 


r  -  X, 


\dx, 


& 


=  o. 


dt2  " v)  dt 

By  contraction  and  multiplication  by  the  scalar  dt  we 
obtain  the  equation  of  kinetic  energy 

,2\ 


mq ‘ 


=  Xvdxv. 


If  denotes  the  difference  of  the  co-ordinates  of 
the  material  particle  and  a  point  fixed  in  space,  then 
the  %v  have  the  character  of  vectors.  We  evidently 


have 


d2xv  d2^v 


dt2  ~  ~dd'  SO  ^at  e9ua^10ns  °f  m°ti°n  of  the 
particle  may  be  written 

Multiplying  this  equation  by  f  we  obtain  a  tensor 
equation 


(>*w  - 


Contracting  the  tensor  on  the  left  and  taking  the  time 
average  we  obtain  the  virial  theorem,  which  we  shall 
not  consider  further.  By  interchanging  the  indices  and 
subsequent  subtraction,  we  obtain,  after  a  simple  trans¬ 
formation,  the  theorem  of  moments, 


It  is  evident  in  this  way  that  the  moment  of  a  vector 


20  THE  MEANING  OF  RELATIVITY 


is  not  a  vector  but  a  tensor.  On  account  of  their  skew- 
symmetrical  character  there  are  not  nine,  but  only  three 
independent  equations  of  this  system.  The  possibility  of 
replacing  skew-symmetrical  tensors  of  the  second  rank  in 
space  of  three  dimensions  by  vectors  depends  upon  the 
formation  of  the  vector 

A  —  -  A  8 

[i.  CTT^CTTfl. 

If  we  multiply  the  skew- symmetrical  tensor  of  rank  2 
by  the  special  skew-symmetrical  tensor  8  introduced 
above,  and  contract  twice,  a  vector  results  whose  compon¬ 
ents  are  numerically  equal  to  those  of  the  tensor.  These 
are  the  so-called  axial  vectors  which  transform  differ¬ 
ently,  from  a  right-handed  system  to  a  left-handed  system, 
from  the  There  is  a  gain  in  picturesqueness  in 

regarding  a  skew-symmetrical  tensor  of  rank  2  as  a  vector 
in  space  of  three  dimensions,  but  it  does  not  represent 
the  exact  nature  of  the  corresponding  quantity  so  well  as 
considering  it  a  tensor. 

We  consider  next  the  equations  of  motion  of  a  con¬ 
tinuous  medium.  Let  p  be  the  density,  uv  the  velocity 
components  considered  as  functions  of  the  co-ordinates  and 
the  time,  Xv  the  volume  forces  per  unit  of  mass,  and  pva 
the  stresses  upon  a  surface  perpendicular  to  the  c-axis 
in  the  direction  of  increasing  xv.  Then  the  equations  of 
motion  are,  by  Newton’s  law, 

~^Pvcr  -yjr 

PHi  =  "  55“  +  Px* 

in  which  is  the  acceleration  of  the  particle  which  at 


PRE-RELATIVITY  PHYSICS  21 


time  t  has  the  co-ordinates  x*  If  we  express  this 
acceleration  by  partial  differential  coefficients,  we  obtain, 
after  dividing  by  p , 


"duv 

1st 


+ 


l 

P 


+  Xv 


(i  6) 


We  must  show  that  this  equation  holds  independently 
of  the  special  choice  of  the  Cartesian  system  of  co-ordinates. 

lsuv  'buv  . 

(«„)  is  a  vector,  and  therefore  -r—  is  also  a  vector,  r —  is 


a  tensor  of  rank  2,  ^~^uT  is  a  tensor  of  rank  3.  The  second 

OXg- 

term  on  the  left  results  from  contraction  in  the  indices 
cr,  r.  The  vector  character  of  the  second  term  on  the  right 
is  obvious.  In  order  that  the  first  term  on  the  right  may 
also  be  a  vector  it  is  necessary  for  pv(J  to  be  a  tensor. 


^Pl<T 

Then  by  differentiation  and  contraction  r— ^  results,  and 

0*0- 

is  therefore  a  vector,  as  it  also  is  after  multiplication  by 


the  reciprocal  scalar  -  •  That pv(T  is  a  tensor,  and  therefore 
transforms  according  to  the  equation 


P  =  ^na^vfipafl  ) 

is  proved  in  mechanics  by  integrating  this  equation  over 
an  infinitely  small  tetrahedron.  It  is  also  proved  there 
by  application  of  the  theorem  of  moments  to  an  infinitely 
small  parallelopipedon,  that  pv(J  =  pav)  and  hence  that  the 
tensor  of  the  stress  is  a  symmetrical  tensor.  From  what 
has  been  said  it  follows  that,  with  the  aid  of  the  rules 


22  THE  MEANING  OF  RELATIVITY 


given  above,  the  equation  is  co-variant  with  respect  to 
orthogonal  transformations  in  space  (rotational  trans¬ 
formations)  ;  and  the  rules  according  to  which  the 
quantities  in  the  equation  must  be  transformed  in  order 
that  the  equation  may  be  co-variant  also  become  evident. 

The  co-variance  of  the  equation  of  continuity, 


tp  Xpuv) 

37 + 


requires,  from  the  foregoing,  no  particular  discussion. 

We  shall  also  test  for  co-variance  the  equations  which 
express  the  dependence  of  the  stress  components  upon 
the  properties  of  the  matter,  and  set  up  these  equations 
for  the  case  of  a  compressible  viscous  fluid  with  the  aid 
of  the  conditions  of  co-variance.  If  we  neglect  the  vis¬ 
cosity,  the  pressure,  />,  will  be  a  scalar,  and  will  depend 
only  upon  the  density  and  the  temperature  of  the  fluid. 
The  contribution  to  the  stress  tensor  is  then  evidently 

pK* 


in  which  is  the  special  symmetrical  tensor.  This  term 
will  also  be  present  in  the  case  of  a  viscous  fluid.  But  in 
this  case  there  will  also  be  pressure  terms,  which  depend 
upon  the  space  derivatives  of  the  uv .  We  shall  assume 
that  this  dependence  is  a  linear  one.  Since  these  terms 
must  be  symmetrical  tensors,  the  only  ones  which  enter 
will  be 


(for  r  *  is  a  scalar).  For  physical  reasons  (no  slipping) 

a 


PRE  RELATIVITY  PHYSICS  23 


it  is  assumed  that  for  symmetrical  dilatations  in  all 
directions,  i.e.  when 


bu2  bu3  bUj 
'bx1  ~  bx2  ~  bx2  ’  bx2 


,  etc.,  =  o, 


bx2  bx3  ’  bx2 
there  are  no  frictional  forces  present,  from  which  it 

2  1  bu, 

follows  that  /3  =  -  -a.  If  only  ^7  is  different  from 


bu1 


zero,  let  p3l  =  -  77- — ,  by  which  a  is  determined.  We 


then  obtain  for  the  complete  stress  tensor, 

rY^u  .  ^y\  2{bU,  bu2  bu3\  *  “1  , 

(*s) 

The  heuristic  value  of  the  theory  of  invariants,  which 
arises  from  the  isotropy  of  space  (equivalence  of  all 
directions),  becomes  evident  from  this  example. 

We  consider,  finally,  Maxwell’s  equations  in  the  form 
which  are  the  foundation  of  the  electron  theory  of  Lorentz. 


u3 

I 

bex 

+ 

I 

dx2 

^x3 

C 

bt 

c 

I 

be 2 

+ 

1 

^3 

Da*! 

c 

bi 

c 

• 

be. 

• 

be. 

• 

Tie. 

1  + 

+ 

0  _ 

p 

bx1 

bx.2 

bx 

3 

r 

^3  _ 

'be2 

I  bhx 

^2 

bx2 

c 

Tit 

be1 

^■3 

I  bh2 

^3 

bxl 

c 

bt 

(19) 


•  (20) 


24  THE  MEANING  OF  RELATIVITY 

i  is  a  vector,  because  the  current  density  is  defined  as 
the  density  of  electricity  multiplied  by  the  vector  velocity 
of  the  electricity.  According  to  the  first  three  equations 
it  is  evident  that  e  is  also  to  be  regarded  as  a  vector. 
Then  h  cannot  be  regarded  as  a  vector.*  The  equations 
may,  however,  easily  be  interpreted  if  h  is  regarded  as  a 
symmetrical  tensor  of  the  second  rank.  In  this  sense,  we 
write  /z23,  k31)  ^12j  in  place  of  hx,  k2i  hz  respectively.  Pay¬ 
ing  attention  to  the  skew-symmetry  of  k^,  the  first  three 
equations  of  (19)  and  (20)  may  be  written  in  the  form 


_  1  +  L  { 

bxv  C  bt  C  11 

'K.  -  ^  =  +  I 

bxv  bx^  C  bt 


(19a) 

(20a) 


In  contrast  to  e,  h  appears  as  a  quantity  which  has  the 
same  type  of  symmetry  as  an  angular  velocity.  The 
divergence  equations  then  take  the  form 


=  P  .  .  .  (i9b) 

+  'bhyp  bh?iX  _  o 

bxp  bx^  bxv 

The  last  equation  is  a  skew-symmetrical  tensor  equation 
of  the  third  rank  (the  skew-symmetry  of  the  left-hand 
side  with  respect  to  every  pair  of  indices  may  easily  be 


*  These  considerations  will  make  the  reader  familiar  with  tensor  opera¬ 
tions  without  the  special  difficulties  of  the  four-dimensional  treatment; 
corresponding  considerations  in  the  theory  of  special  relativity  (Minkowski’s* 
interpretation  of  the  field)  will  then  offer  fewer  difficulties, 


PRE-RELATIVITY  PHYSICS  25 

* 

1 

proved,  if  attention  is  paid  to  the  skew-symmetry  cf  k^). 
This  notation  is  more  natural  than  the  usual  one,  because, 
in  contrast  to  the  latter,  it  is  applicable  to  Cartesian  left- 
handed  systems  as  well  as  to  right-handed  systems  without 
change  of  sign. 


LECTURE  II 


THE  THEORY  OF  SPECIAL  RELATIVITY 

THE  previous  considerations  concerning  the  configura¬ 
tion  of  rigid  bodies  have  been  founded,  irrespective 
of  the  assumption  as  to  the  validity  of  the  Euclidean 
geometry,  upon  the  hypothesis  that  all  directions  in  space, 
or  all  configurations  of  Cartesian  systems  of  co-ordinates, 
are  physically  equivalent.  We  may  express  this  as  the 
“  principle  of  relativity  with  respect  to  direction,”  and  it 
has  been  shown  how  equations  (laws  of  nature)  may  be 
found,  in  accord  with  this  principle,  by  the  aid  of  the 
calculus  of  tensors.  We  now  inquire  whether  there  is  a 
relativity  with  respect  to  the  state  of  motion  of  the  space 
of  reference ;  in  other  words,  whether  there  are  spaces  of 
reference  in  motion  relatively  to  each  other  which  are 
physically  equivalent.  From  the  standpoint  of  mechanics 
it  appears  that  equivalent  spaces  of  reference  do  exist. 
For  experiments  upon  the  earth  tell  us  nothing  of  the 
fact  that  we  are  moving  about  the  sun  with  a  velocity  of 
approximately  30  kilometres  a  second.  On  the  other 
hand,  this  physical  equivalence  does  not  seem  to  hold  for 
spaces  of  reference  in  arbitrary  motion ;  for  mechanical 
effects  do  not  seem  to  be  subject  to  the  same  laws  in  a 
jolting  railway  train  as  in  one  moving  with  uniform 

26 


SPECIAL  RELATIVITY 


27 


velocity ;  the  rotation  of  the  earth  must  be  considered  in 
writing  down  the  equations  of  motion  relatively  to  the 
earth.  It  appears,  therefore,  as  if  there  were  Cartesian 
systems  of  co-ordinates,  the  so-called  inertial  systems,  with 
reference  to  which  the  laws  of  mechanics  (more  generally 
the  laws  of  physics)  are  expressed  in  the  simplest  form. 
We  may  infer  the  validity  of  the  following  theorem  :  If 
K  is  an  inertial  system,  then  every  other  system  K'  which 
moves  uniformly  and  without  rotation  relatively  to  K ,  is 
also  an  inertial  system ;  the  laws  of  nature  are  in  con¬ 
cordance  for  all  inertial  systems.  This  statement  we  shall 
call  the  “  principle  of  special  relativity.”  We  shall  draw 
certain  conclusions  from  this  principle  of  “  relativity  of 
translation  ”  just  as  we  have  already  done  for  relativity  of 
direction. 

In  order  to  be  able  to  do  this,  we  must  first  solve  the 
following  problem.  If  we  are  given  the  Cartesian  co¬ 
ordinates,^,  and  the  time  /,  of  an  event  relatively  to  one 
inertial  system,  K ,  how  can  we  calculate  the  co-ordinates, 
x v ,  and  the  time,  of  the  same  event  relatively  to  an 
inertial  system  K'  which  moves  with  uniform  trans¬ 
lation  relatively  to  K  ?  In  the  pre-relativity  physics 
this  problem  was  solved  by  making  unconsciously  two 
hypotheses  : — 

i.  The  time  is  absolute;  the  time  of  an  event,  t\ 
relatively  to  K'  is  the  same  as  the  time  relatively  to  K. 
If  instantaneous  signals  could  be  sent  to  a  distance,  and 
if  one  knew  that  the  state  of  motion  of  a  clock  had  no 
influence  on  its  rate,  then  this  assumption  would  be 
physically  established.  For  then  clocks,  similar  to  one 


28  THE  MEANING  OF  RELATIVITY 


another,  and  regulated  alike,  could  be  distributed  over 
the  systems  K  and  K\  at  rest  relatively  to  them,  and 
their  indications  would  be  independent  of  the  state  of 
motion  of  the  systems  ;  the  time  of  an  event  would  then 
be  given  by  the  clock  in  its  immediate  neighbourhood. 

2.  Length  is  absolute  ;  if  an  interval,  at  rest  relatively 
to  K,  has  a  length  s,  then  it  has  the  same  length  s, 
relatively  to  a  system  K'  which  is  in  motion  relatively 
to  K. 

If  the  axes  of  K  and  K'  are  parallel  to  each  other,  a 
simple  calculation  based  on  these  two  assumptions,  gives 
the  equations  of  transformation 

xv  =  xv  -  av  -  bvt 
t'  =  t  -  b 

This  transformation  is  known  as  the  “  Galilean  Trans¬ 
formation.”  Differentiating  twice  by  the  time,  we  get 

d2x  v  d2xv 

~dF  =  ~dFm 


Further,  it  follows  that  for  two  simultaneous  events, 


J  a)  _ 


x 


(2)  =  ^  (1)  _ 


(2) 


The  invariance  of  the  distance  between  the  two  points 
results  from  squaring  and  adding.  From  this  easily 
follows  the  co-variance  of  Newton’s  equations  of  motion 
with  respect  to  the  Galilean  transformation  (21).  Hence 
it  follows  that  classical  mechanics  is  in  accord  with  the 
principle  of  special  relativity  if  the  two  hypotheses 
respecting  scales  and  clocks  are  made. 

But  this  attempt  to  found  relativity  of  translation  upon 
the  Galilean  transformation  fails  when  applied  to  electron 


SPECIAL  RELATIVITY 


29 


magnetic  phenomena.  The  Maxwell-Lorentz  electro¬ 
magnetic  equations  are  not  co-variant  with  respect  to  the 
Galilean  transformation.  In  particular,  we  note,  by  (21), 
that  a  ray  of  light  which  referred  to  K  has  a  velocity  c, 
has  a  different  velocity  referred  to  K\  depending  upon 
its  direction.  The  space  of  reference  of  K  is  therefore 
distinguished,  with  respect  to  its  physical  properties,  from 
all  spaces  of  reference  which  are  in  motion  relatively  to  it 
(quiescent  sether).  But  all  experiments  have  shown  that 
electro-magnetic  and  optical  phenomena,  relatively  to  the 
earth  as  the  body  of  reference,  are  not  influenced  by  the 
translational  velocity  of  the  earth.  The  most  important 
of  these  experiments  are  those  of  Michelson  and  Morley, 
which  I  shall  assume  are  known.  The  validity  of  the 
principle  of  special  relativity  can  therefore  hardly  be 
doubted. 

O11  the  other  hand,  the  Maxwell-Lorentz  equations 
have  proved  their  validity  in  the  treatment  of  optical 
problems  in  moving  bodies.  No  other  theory  has 
satisfactorily  explained  the  facts  of  aberration,  the 
propagation  of  light  in  moving  bodies  (Flzeau),  and 
phenomena  observed  in  double  stars  (De  Sitter).  The 
consequence  of  the  Maxwell-Lorentz  equations  that  in  a 
vacuum  light  is  propagated  with  the  velocity  c,  at  least 
with  respect  to  a  definite  inertial  system  K,  must  there¬ 
fore  be  regarded  as  proved.  According  to  the  principle 
of  special  relativity,  we  must  also  assume  the  truth  of 
this  principle  for  every  other  inertial  system. 

Before  we  draw  any  conclusions  from  these  two 
principles  we  must  first  review  the  physical  significance 


30  THE  MEANING  OF  RELATIVITY 

of  the  concepts  “time”  and  “velocity.”  It  follows  from 
what  has  gone  before,  that  co-ordinates  with  respect  to 
an  inertial  system  are  physically  defined  by  means  of 
measurements  and  constructions  with  the  aid  of  rigid 
bodies.  In  order  to  measure  time,  we  have  supposed  a 
clock,  Uy  present  somewhere,  at  rest  relatively  to  K.  But 
we  cannot  fix  the  time,  by  means  of  this  clock,  of  an  event 
whose  distance  from  the  clock  is  not  negligible  ;  for  there 
are  no  “  instantaneous  signals  ”  that  we  can  use  in  order 
to  compare  the  time  of  the  event  with  that  of  the  clock. 
In  order  to  complete  the  definition  of  time  we  may 
employ  the  principle  of  the  constancy  of  the  velocity  of 
light  in  a  vacuum.  Let  us  suppose  that  we  place  similar 
clocks  at  points  of  the  system  K ,  at  rest  relatively  to  it, 
and  regulated  according  to  the  following  scheme.  A  ray 
of  light  is  sent  out  from  one  of  the  clocks,  Um,  at  the 
instant  when  it  indicates  the  time  tm)  and  travels  through 
a  vacuum  a  distance  rmn}  to  the  clock  Un  ;  at  the  instant 
when  this  ray  meets  the  clock  Un  the  latter  is  set  to 

indicate  the  time  tn  =  tm  4-  —  .*  The  principle  of  the 

c 

constancy  of  the  velocity  of  light  then  states  that  this 
adjustment  of  the  clocks  wall  not  lead  to  contradictions. 
With  clocks  so  adjusted,  we  can  assign  the  time  to  events 
which  take  place  near  any  one  of  them.  It  is  essential  to 

*  Strictly  speaking,  it  would  be  more  correct  to  define  simultaneity  first, 
somewhat  as  follows :  two  events  taking  place  at  the  points  A  and  B  of 
the  system  K  are  simultaneous  if  they  appear  at  the  same  instant  when 
observed  from  the  middle  point,  M,  of  the  interval  AB.  Time  is  then 
defined  as  the  ensemble  of  the  indications  of  similar  clocks,  at  rest 
relatively  to  K,  which  register  the  same  simultaneously. 


SPECIAL  RELATIVITY  31 

note  that  this  definition  of  time  relates  only  to  the  inertial 
system  K ,  since  we  have  used  a  system  of  clocks  at  rest 
relatively  to  K.  The  assumption  which  was  made  in  the 
pre-relativity  physics  of  the  absolute  character  of  time 
^i.e.  the  independence  of  time  of  the  choice  of  the  inertial 
system)  does  not  follow  at  all  from  this  definition. 

The  theory  of  relativity  is  often  criticized  for  giving, 
without  justification,  a  central  theoretical  role  to  the 
propagation  of  light,  in  that  it  founds  the  concept  of  time 
upon  the  law  of  propagation  of  light.  The  situation, 
however,  is  somewhat  as  follows.  In  order  to  give 
physical  significance  to  the  concept  of  time,  processes  of 
some  kind  are  required  which  enable  relations  to  be 
established  between  different  places.  It  is  immaterial 
what  kind  of  processes  one  chooses  for  such  a  definition 
of  time.  It  is  advantageous,  however,  for  the  theory, 
to  choose  only  those  processes  concerning  which  we  know 
something  certain.  This  holds  for  the  propagation  of 
light  in  vacuo  in  a  higher  degree  than  for  any  other  process 
which  could  be  considered,  thanks  to  the  investigations 
of  Maxwell  and  H.  A.  Lorentz. 

From  all  of  these  considerations,  space  and  time  data 
have  a  physically  real,  and  not  a  mere  fictitious,  signifi¬ 
cance  ;  in  particular  this  holds  for  all  the  relations  in 
which  co-ordinates  and  time  enter,  e.g.  the  relations 
(21).  There  is,  therefore,  sense  in  asking  whether  those 
equations  are  true  or  not,  as  well  as  in  asking  what  the 
true  equations  of  transformation  are  by  which  we  pass 
from  one  inertial  system  K  to  another,  K\  moving 
relatively  to  it.  It  may  be  shown  that  this  is  uniquely 


32  THE  MEANING  OF  RELATIVITY 


settled  by  means  of  the  principle  of  the  constancy  of  the 
velocity  of  light  and  the  principle  of  special  relativity. 

To  this  end  we  think  of  space  and  time  physically 
defined  with  respect  to  two  inertial  systems,  K  and  K\  in 
the  way  that  has  been  shown.  Further,  let  a  ray  of  light 
pass  from  one  point  P1  to  another  point  P2  of  K  through 
a  vacuum.  If  r  is  the  measured  distance  between  the  two 
points,  then  the  propagation  of  light  must  satisfy  the 
equation 

r  =  c .  At 

If  we  square  this  equation,  and  express  r 2  by  the 
differences  of  the  co-ordinates,  Axv,  in  place  of  this  equation 
we  can  write 

(A;rv)2  -  c2A  t2  =  o  .  .  (22) 

This  equation  formulates  the  principle  of  the  constancy 
of  the  velocity  of  light  relatively  to  K.  It  must  hold 
whatever  may  be  the  motion  of  the  source  which  emits 
the  ray  of  light. 

The  same  propagation  of  light  may  also  be  considered 
relatively  to  K\  in  which  case  also  the  principle  of  the 
constancy  of  the  velocity  of  light  must  be  satisfied. 
Therefore,  with  respect  to  K',  we  have  the  equation 

^>(ATV)2  -  c2 A/2  =  o  .  (22a) 

Equations  (22a)  and  (22)  must  be  mutually  consistent 
with  each  other  with  respect  to  the  transformation  which 
transforms  from  K  to  K\  A  transformation  which  effects 
this  we  shall  call  a  “Lorentz  transformation.” 

Before  considering  these  transformations  in  detail  we 


SPECIAL  RELATIVITY 


33 


shall  make  a  few  general  remarks  about  space  and  time. 
In  the  pre-relativity  physics  space  and  time  were  separ¬ 
ate  entities.  Specifications  of  time  were  independent  of 
the  choice  of  the  space  of  reference.  The  Newtonian 
mechanics  was  relative  with  respect  to  the  space  of 
reference,  so  that,  e.g.  the  statement  that  two  non-simul- 
taneous  events  happened  at  the  same  place  had  no  objective 
meaning  (that  is,  independent  of  the  space  of  reference). 
But  this  relativity  had  no  role  in  building  up  the  theory. 
One  spoke  of  points  of  space,  as  of  instants  of  time,  as  if 
they  were  absolute  realities.  It  was  not  observed  that 
the  true  element  of  the  space-time  specification  was  the 
event,  specified  by  the  four  numbers  xl}  x2,  xz ,  t.  The 
conception  of  something  happening  was  always  that  of  a 
four-dimensional  continuum  ;  but  the  recognition  of  this 
was  obscured  by  the  absolute  character  of  the  pre-relativity 
time.  Upon  giving  up  the  hypothesis  of  the  absolute 
character  of  time,  particularly  that  of  simultaneity,  the 
four-dimensionality  of  the  time-space  concept  was  im¬ 
mediately  recognized.  It  is  neither  the  point  in  space, 
nor  the  instant  in  time,  at  which  something  happens  that 
has  physical  reality,  but  only  the  event  itself.  There  is 
no  absolute  (independent  of  the  space  of  reference)  relation 
in  space,  and  no  absolute  relation  in  time  between  two 
events,  but  there  is  an  absolute  (independent  of  the  space 
of  reference)  relation  in  space  and  time,  as  will  appear  in 
the  sequel.  The  circumstance  that  there  is  no  objective 
rational  division  of  the  four-dimensional  continuum  into 
a  three-dimensional  space  and  a  one-dimensional  time 
continuum  indicates  that  the  laws  of  nature  will  assume 
3 


34  THE  MEANING  OF  RELATIVITY 


a  form  which  is  logically  most  satisfactory  when  expressed 
as  laws  in  the  four-dimensional  space-time  continuum. 
Upon  this  depends  the  great  advance  in  method  which 
the  theory  of  relativity  owes  to  Minkowski.  Considered 
from  this  standpoint,  we  must  regard  xv  x2,  x3)  t  as  the 
four  co-ordinates  of  an  event  in  the  four-dimensional  con¬ 
tinuum.  We  have  far  less  success  in  picturing  to  ourselves 
relations  in  this  four-dimensional  continuum  than  in  the 
three-dimensional  Euclidean  continuum  ;  but  it  must  be 
emphasized  that  even  in  the  Euclidean  three-dimensional 
geometry  its  concepts  and  relations  are  only  of  an  abstract 
nature  in  our  minds,  and  are  not  at  all  identical  with  the 
images  we  form  visually  and  through  our  sense  of  touch. 
The  non-divisibility  of  the  four-dimensional  continuum 
of  events  does  not  at  all,  however,  involve  the  equivalence 
of  the  space  co-ordinates  with  the  time  co-ordinate.  On 
the  contrary,  we  must  remember  that  the  time  co-ordinate 
is  defined  physically  wholly  differently  from  the  space 
co-ordinates.  The  relations  (22)  and  (22a)  which  when 
equated  define  the  Lorentz  transformation  show,  further, 
a  difference  in  the  role  of  the  time  co-ordinate  from  that 
of  the  space  co-ordinates  ;  for  the  term  At2  has  the  opposite 
sign  to  the  space  terms,  Ax2,  Ax22,  Ax32. 

Before  we  analyse  further  the  conditions  which  define 
the  Lorentz  transformation,  we  shall  introduce  the  light¬ 
time,  l  =  ct ,  in  place  of  the  time,  t,  in  order  that  the 
constant  c  shall  not  enter  explicitly  into  the  formulas  to 
be  developed  later.  Then  the  Lorentz  transformation  is 
defined  in  such  a  way  that,  first,  it  makes  the  equation 

Ax 2  +  Ax2  +  Ax3  -  A l2  =  o  .  (22b) 


SPECIAL  RELATIVITY 


35 


a  co-variant  equation,  that  is,  an  equation  which  is  satisfied 
with  respect  to  every  inertial  system  if  it  is  satisfied  in 
the  inertial  system  to  which  we  refer  the  two  given  events 
(emission  and  reception  of  the  ray  of  light).  Finally, 
with  Minkowski,  we  introduce  in  place  of  the  real  time 
co-ordinate  /  =  ct,  the  imaginary  time  co-ordinate 

=  il  =  ict  -  I  =  z). 

Then  the  equation  defining  the  propagation  of  light, 
which  must  be  co-variant  with  respect  to  the  Lorentz 
transformation,  becomes 

}Ax2  =  A;tq2  +  A^22  +  A;r32  +  A^42  =  o  (22c) 
(4) 

This  condition  is  always  satisfied  *  if  we  satisfy  the  more 
general  condition  that 

s2  =  A^!2  +  A^22  +  A^32  +  A^42  .  (23) 

shall  be  an  invariant  with  respect  to  the  transformation. 
This  condition  is  satisfied  only  by  linear  transformations, 
that  is,  transformations  of  the  type 

■  ■  ■  (24) 

in  which  the  summation  over  the  a  is  to  be  extended 
from  a  =  I  to  a  =  4.  A  glance  at  equations  (23)  and 
(24)  shows  that  the  Lorentz  transformation  so  defined  is 
identical  with  the  translational  and  rotational  transforma¬ 
tions  of  the  Euclidean  geometry,  if  we  disregard  the 
number  of  dimensions  and  the  relations  of  reality.  We 

*  That  this  specialization  lies  in  the  nature  of  the  case  will  be  evident 
later. 


36  THE  MEANING  OF  RELATIVITY 


can  also  conclude  that  the  coefficients  b Ma  must  satisfy  the 
conditions 


^ \iafy va  iv  ^  av 


.  (25) 

Since  the  ratios  of  the  xv  are  real,  it  follows  that  all  the 
a, a  and  the  b Ma  are  real,  except  biV  b±2,  b±3,  bw  bu,  and 
£34,  which  are  purely  imaginary. 

Special  Lorentz  Transformation.  We  obtain  the 
simplest  transformations  of  the  type  of  (24)  and  (25)  if 
only  two  of  the  co-ordinates  are  to  be  transformed,  and  if 
all  the  which  determine  the  new  origin,  vanish.  We 
obtain  then  for  the  indices  1  and  2,  on  account  of  the 
three  independent  conditions  which  the  relations  (25) 
furnish, 

x\  —  xY  cos  cf)  -  x 2  sin  <£ 
x\  =  xx  sin  <f>  +  x2  cos  <jf> 


x3  =  x3 

x\  = 


(26) 


This  is  a  simple  rotation  in  space  of  the  (space) 
co-ordinate  system  about  ^3-axis.  We  see  that  the 
rotational  transformation  in  space  (without  the  time 
transformation)  which  we  studied  before  is  contained  in 
the  Lorentz  transformation  as  a  special  case.  For  the 
indices  1  and  4  we  obtain,  in  an  analogous  manner, 


x\  =  x1  cos  \fr  -  x±  sin  yfr 
x\  —  xx  sin  y/r  +  x4  cos  yjr 

X  o  =a  X  n 


X  3  =  *3 


I 


(26a) 


On  account  of  the  relations  of  reality  yjr  must  be  taken 
as  imaginary.  To  interpret  these  equations  physically, 
we  introduce  the  real  light-time  l  and  the  velocity  v  of 


SPECIAL  RELATIVITY 


37 


K'  relatively  to  K ,  instead  of  the  imaginary  angle  yjr.  We 
have,  first, 

x\  =  x\  cos  ^  _  sin  yjr 
l  —  -  MTj  sin  yjr  +  /  cos 


Since  for  the  origin  of  K\  i.e.,  for  xx  =  o,  we  must  have 
Aq  ==  it  follows  from  the  first  of  these  equations  that 


and  also 


so  that  we  obtain 


v  =  i  tan  -v/r 


sin  y\r  = 


-  iv 


s/l  -  V 1 


COS  yjr  =  7= 


F2/ 


-T ,  = 


/'  = 


xx  -  vl  ^ 

Jl  -  z/2 

/  -  ZUq 

v/T  “  v  ‘2 


X2 

Xo 


=  ^ 


(27) 

(28) 


(29) 


These  equations  form  the  well-known  special  Lorentz 
transformation,  which  in  the  general  theory  represents  a 
rotation,  through  an  imaginary  angle,  of  the  four-dimen¬ 
sional  system  of  co-ordinates.  If  we  introduce  the  ordinary 
time  t,  in  place  of  the  light-time  /,  then  in  (29)  we  must 

v 

replace  l  by  ct  and  v  by  -• 

We  must  now  fill  in  a  gap.  From  the  principle  of  the 
constancy  of  the  velocity  of  light  it  follows  that  the 
equation 

A.V2  =  o 


38  THE  MEANING  OF  RELATIVITY 

has  a  significance  which  is  independent  of  the  choice  of 
the  inertial  system  ;  but  the  invariance  of  the  quantity 

does  not  at  all  follow  from  this.  This  quantity 

might  be  transformed  with  a  factor.  This  depends  upon 
the  fact  that  the  right-hand  side  of  (29)  might  be  multi¬ 
plied  by  a  factor  independent  of  v.  But  the  principle 
of  relativity  does  not  permit  this  factor  to  be  different  from 
1,  as  we  shall  now  show.  Let  us  assume  that  we  have 
a  rigid  circular  cylinder  moving  in  the  direction  of  its 
axis.  If  its  radius,  measured  at  rest  with  a  unit  measur¬ 
ing  rod  is  equal  to  R0i  its  radius  R  in  motion,  might  be 
different  from  R0,  since  the  theory  of  relativity  does  not 
make  the  assumption  that  the  shape  of  bodies  with  respect 
to  a  space  of  reference  is  independent  of  their  motion 
relatively  to  this  space  of  reference.  But  all  directions 
in  space  must  be  equivalent  to  each  other.  R  may  there¬ 
fore  depend  upon  the  magnitude  q  of  the  velocity,  but 
not  upon  its  direction ;  R  must  therefore  be  an  even 
function  of  q.  If  the  cylinder  is  at  rest  relatively  to  K' 
the  equation  of  its  lateral  surface  is 

;r'2  +  /2  =  R02. 

If  we  write  the  last  two  equations  of  (29)  more  generally 


then  the  lateral  surface  of  the  cylinder  referred  to  K 
satisfies  the  equation 

R  2 
-^0 


SPECIAL  RELATIVITY 


39 


The  factor  X  therefore  measures  the  lateral  contraction  of 
the  cylinder,  and  can  thus,  from  the  above,  be  only  an 
even  function  of  v. 

If  we  introduce  a  third  system  of  co-ordinates,  K" , 
which  moves  relatively  to  K'  with  velocity  v  in  the  direc¬ 
tion  of  the  negative  ^r-axis  of  K,  we  obtain,  by  apply¬ 
ing  (29)  twice, 

x\  —  X(z;)X(  -  v)x± 


•  •  •  • 

/"  =  X(v)X(  -  v)l. 

Now,  since  \(v)  must  be  equal  to  X(  -  v ),  and  since  we 
assume  that  we  use  the  same  measuring  rods  in  all  the 
systems,  it  follows  that  the  transformation  of  K"  to  K 
must  be  the  identical  transformation  (since  the  possibility 
X  =  —  I  does  not  need  to  be  considered).  It  is  essential 
for  these  considerations  to  assume  that  the  behaviour  of 
the  measuring  rods  does  not  depend  upon  the  history  of 
their  previous  motion. 

Moving  Measuring  Rods  and  Clocks.  At  the  definite  K- 
time,  l=o,  the  position  of  the  points  given  by  the  integers 
x\  =  n,  is  with  respect  to  K,  given  by  xx  =  n yj  1  -  v2, ; 
this  follows  from  the  first  of  equations  (29)  and  expresses 
the  Lorentz  contraction.  A  clock  at  rest  at  the  origin 
xY  =  o  of  K ,  whose  beats  are  characterized  by  /  =  n,  will, 
when  observed  from  K',  have  beats  characterized  by 

n 

1  =  Vr=7^; 

this  follows  from  the  second  of  equations  (29)  and  shows 


40  THE  MEANING  OF  RELATIVITY 


that  the  clock  goes  slower  than  if  it  were  at  rest  relatively 
to  K' .  These  two  consequences,  which  hold,  mutatis 
mutandis ,  for  every  system  of  reference,  form  the  physical 
content,  free  from  convention,  of  the  Lorentz  transforma¬ 
tion. 

Addition  Theorem  for  Velocities.  If  we  combine  two 
special  Lorentz  transformations  with  the  relative  velocities 
v1  and  v2,  then  the  velocity  of  the  single  Lorentz  trans¬ 
formation  which  takes  the  place  of  the  two  separate  ones 
is,  according  to  (27),  given  by 


vu  = 1 


,  ,  ,  N  .  tan  'v k  +  tan  aK 

tan  (*  x  +  _  t;n  ^  tan\  = 


V,  +  V, 


2  1  +  ^2  ’ 


(30) 


General  Statements  about  the  Lorentz  Transformation 
and  its  Theory  of  Invariants.  The  whole  theory  of 
invariants  of  the  special  theory  of  relativity  depends  upon 
the  invariant  a2  (23).  Formally,  it  has  the  same  role  in 
the  four-dimensional  space-time  continuum  as  the  in¬ 
variant  A-vp  +  Arq2  +  A^32  in  the  Euclidean  geometry 
and  in  the  pre-relativity  physics.  The  latter  quantity  is 
not  an  invariant  with  respect  to  all  the  Lorentz  transfor¬ 
mations ;  the  quantity  a2  of  equation  (23)  assumes  the 
role  of  this  invariant.  With  respect  to  an  arbitrary 
inertial  system,  a2  may  be  determined  by  measurements  ; 
with  a  given  unit  of  measure  it  is  a  completely  determinate 
quantity,  associated  with  an  arbitrary  pair  of  events. 

The  invariant  a2  differs,  disregarding  the  number  of 
dimensions,  from  the  corresponding  invariant  of  the 
Euclidean  geometry  in  the  following  points.  In  the 
Euclidean  geometry  a2  is  necessarily  positive  ;  it  vanishes 


SPECIAL  RELATIVITY 


41 


only  when  the  two  points  concerned  come  together.  On 
the  other  hand,  from  the  vanishing  of 


s2  =  ^Aav  =  A-t'f  +  Aal2  +  Aa'32  -  A  t2 

l 


it  cannot  be  concluded  that  the  two  space-time  points 
fall  together;  the  vanishing  of  this  quantity  s2,  is  the 
invariant  condition  that  the  two  space-time  points  can  be 
connected  by  a  light  signal  in  vacuo.  If  P  is  a  point 


42  THE  MEANING  OF  RELATIVITY 


(event)  represented  in  the  four-dimensional  space  of  the 
xv  x2i  x3)  /,  then  all  the  “  points  ”  which  can  be  connected 
to  P  by  means  of  a  light  signal  lie  upon  the  cone  s2  —  o 
(compare  Fig.  I,  in  which  the  dimension  x3  is  suppressed). 
The  “  upper  ”  half  of  the  cone  may  contain  the  “  points  ” 
to  which  light  signals  can  be  sent  from  P;  then  the 
“  lower  ”  half  of  the  cone  will  contain  the  “  points  ”  from 
which  light  signals  can  be  sent  to  P.  The  points  P' 
enclosed  by  the  conical  surface  furnish,  with  P,  a  negative 
s2 ;  PP',  as  well  as  P'P  is  then,  according  to  Minkowski, 
of  the  nature  of  a  time.  Such  intervals  represent  elements 
of  possible  paths  of  motion,  the  velocity  being  less  than 
that  of  light.*  In  this  case  the  /-axis  may  be  drawn  in 
the  direction  of  PP '  by  suitably  choosing  the  state  of 
motion  of  the  inertial  system.  If  P'  lies  outside  of  the 
“light-cone”  then  PP'  is  of  the  nature  of  a  space;  in 
this  case,  by  properly  choosing  the  inertial  system,  A/ 
can  be  made  to  vanish. 

By  the  introduction  of  the  imaginary  time  variable, 
x±  =  z7,  Minkowski  has  made  the  theory  of  invariants  for 
the  four-dimensional  continuum  of  physical  phenomena 
fully  analogous  to  the  theory  of  invariants  for  the  three- 
dimensional  continuum  of  Euclidean  space.  The  theory 
of  four-dimensional  tensors  of  special  relativity  differs  from 
the  theory  of  tensors  in  three-dimensional  space,  therefore, 
only  in  the  number  of  dimensions  and  the  relations  of 
reality. 

*  That  material  velocities  exceeding  that  of  light  are  not  possible, 
follows  from  the  appearance  of  the  radical  i  -  v 2  in  the  special  Lorentz 
transformation  (29). 


SPECIAL  RELATIVITY 


43 


A  physical  entity  which  is  specified  by  four  quantities, 
Av ,  in  an  arbitrary  inertial  system  of  the  xly  x2,  x3,  x±,  is 
called  a  4-vector,  with  the  components  Av,  if  the  Av 
correspond  in  their  relations  of  reality  and  the  properties 
of  transformation  to  the  Axv ;  it  may  be  of  the  nature  of 
a  space  or  of  a  time.  The  sixteen  quantities,  A^v  then 
form  the  components  of  a  tensor  of  the  second  rank,  if 
they  transform  according  to  the  scheme 

A  fiV  * 

It  follows  from  this  that  the  A^v  behave,  with  respect  to 
their  properties  of  transformation  and  their  properties 
of  reality,  as  the  products  of  components,  U^Vv,  of  two 
4-vectors,  (£7)  and  ( V).  All  the  components  are  real 
except  those  which  contain  the  index  4  once,  those  being 
purely  imaginary.  Tensors  of  the  third  and  higher  ranks 
may  be  defined  in  an  analogous  way.  The  operations 
of  addition,  subtraction,  multiplication,  contraction  and 
differentiation  for  these  tensors  are  wholly  analogous  to 
the  corresponding  operations  for  tensors  in  three-dimen¬ 
sional  space. 

Before  we  apply  the  tensor  theory  to  the  four-dimen¬ 
sional  space-time  continuum,  we  shall  examine  more 
particularly  the  skew-symmetrical  tensors.  The  tensor 
of  the  second  rank  has,  in  general,  16  =  4.4  components. 
In  the  case  of  skew-symmetry  the  components  with  two 
equal  indices  vanish,  and  the  components  with  unequal 
indices  are  equal  and  opposite  in  pairs.  There  exist, 
therefore,  only  six  independent  components,  as  is  the 
case  in  the  electromagnetic  field.  In  fact,  it  will  be  shown 


44  THE  MEANING  OF  RELATIVITY 


when  we  consider  Maxwell’s  equations  that  these  may 
be  looked  upon  as  tensor  equations,  provided  we  regard 
the  electromagnetic  field  as  a  skew-symmetrical  tensor. 
Further,  it  is  clear  that  the  skew-symmetrical  tensor  of 
the  third  rank  (skew-symmetrical  in  all  pairs  of  indices) 
has  only  four  independent  components,  since  there  are 
only  four  combinations  of  three  different  indices. 

We  now  turn  to  Maxwell’s  equations  (19a),  (19b),  (20a)> 
(20b),  and  introduce  the  notation  :  * 


023 

^23 

031 

h3i 

012 

hi2 

014 

-  ie* 

024  03il 

-  tey  -  iezj 

■  (30a) 

Ji 

1 

/. 

I 

/a 

I 

i  . 

•  (30 

'cl* 

-  i y 

c 

-  \z 

c  z 

ip\ 

1 

with  the  convention 

that 

</>M„  shall 

be 

equal  to 

*pvix' 

Then  Maxwell’s  equations  may  be  combined  into  the 
forms 


A 

3  ^  iii'  3<Lcr  3  rf-1  u  )L 

tx* 


(32) 

(33) 


as  one  can  easily  verify  by  substituting  from  (30a)  and 
(31).  Equations  (32)  and  (33)  have  a  tensor  character, 
and  are  therefore  co-variant  with  respect  to  Lorentz 
transformations,  if  the  <£MJ,  and  the  J ,x  have  a  tensor 
character,  which  we  assume.  Consequently,  the  laws  for 


*  In  order  to  avoid  confusion  from  now  on  we  shall  use  the  three- 
dimensional  space  indices,  x,  y,  z  instead  of  1,  2,  3,  and  we  shall  reserve 
the  numeral  indices  1,  2,  3,  4  for  the  four-dimensional  space-time  con¬ 
tinuum. 


SPECIAL  RELATIVITY 


45 


transforming  these  quantities  from  one  to  another  allow¬ 
able  (inertial)  system  of  co-ordinates  are  uniquely 
determined.  The  progress  in  method  which  electro¬ 
dynamics  owes  to  the  theory  of  special  relativity  lies 
principally  in  this,  that  the  number  of  independent 
hypotheses  is  diminished.  If  we  consider,  for  example, 
equations  (19a)  only  from  the  standpoint  of  relativity  of 
direction,  as  we  have  done  above,  we  see  that  they  have 
three  logically  independent  terms.  The  way  in  which 
the  electric  intensity  enters  these  equations  appears  to 
be  wholly  independent  of  the  way  in  which  the  magnetic 
intensity  enters  them  ;  it  would  not  be  surprising  if  instead 

c)2e 

of  -^jr,  we  had,  say,  or  if  this  term  were  absent.  On 


the  other  hand,  only  two  independent  terms  appear  in 
equation  (32).  The  electromagnetic  field  appears  as  a 
formal  unit ;  the  way  in  which  the  electric  field  enters 
this  equation  is  determined  by  the  way  in  which  the 
magnetic  field  enters  it.  Besides  the  electromagnetic 
field,  only  the  electric  current  density  appears  as  an 
independent  entity.  This  advance  in  method  arises  from 
the  fact  that  the  electric  and  magnetic  fields  draw  their 
separate  existences  from  the  relativity  of  motion.  A 
field  which  appears  to  be  purely  an  electric  field,  judged 
from  one  system,  has  also  magnetic  field  components 
when  judged  from  another  inertial  system.  When  applied 
to  an  electromagnetic  field,  the  general  law  of  transforma¬ 
tion  furnishes,  for  the  special  case  of  the  special  Lorentz 
transformation,  the  equations 


46  THE  MEANING  OF  RELATIVITY 


X 

h'*  =  h*  -j 

Cy  —  V\\z 

,  ,  K  + 

y  \J  I  -  V2 

>  x/i  -^4 

e*  +  vhy 

,  ,  -  vey 

*/  I  -  V2 

*  v/l  -vJ 

If  there  exists  with  respect  to  K  only  a  magnetic  field, 
h,  but  no  electric  field,  e,  then  with  respect  to  K'  there 
exists  an  electric  field  e'  as  well,  which  would  act  upon 
an  electric  particle  at  rest  relatively  to  K' .  An  observer 
at  rest  relatively  to  K  would  designate  this  force  as  the 
Biot-Savart  force,  or  the  Lorentz  electromotive  force.  It 
therefore  appears  as  if  this  electromotive  force  had  become 
fused  with  the  electric  field  intensity  into  a  single  entity. 

In  order  to  view  this  relation  formally,  let  us  consider 
the  expression  for  the  force  acting  upon  unit  volume  of 
electricity, 

k  =  pe  +  [i,  h]  .  .  .  (35) 

in  which  i  is  the  vector  velocity  of  electricity,  with  the 
velocity  of  light  as  the  unit.  If  we  introduce  and 
according  to  (30a)  and  (31),  we  obtain  for  the  first 
component  the  expression 

$12  J 2  +  ^13^3  + 

Observing  that  <f)n  vanishes  on  account  of  the  skew- 
symmetry  of  the  tensor  (<£),  the  components  of  k  are  given 
by  the  first  three  components  of  the  four-dimensional 
vector 

=  J v  (3^) 

and  the  fourth  component  is  given  by 

+  fy&J 2  fy&J 3  ~  "f"  =  ^  •  (37) 


SPECIAL  RELATIVITY 


47 


There  is,  therefore,  a  four-dimensional  vector  of  force  per 
unit  volume,  whose  first  three  components,  kv  k.2)  k3,  are 
the  ponderomotive  force  components  per  unit  volume,  and 
whose  fourth  component  is  the  rate  of  working  of  the  field 

per  unit  volume,  multiplied  by  ^  -  I. 


A  comparison  of  (36)  and  (35)  shows  that  the  theory 
of  relativity  formally  unites  the  ponderomotive  force  of 
the  electric  field,  pe,  and  the  Biot-Savart  or  Lorentz 
force  [i,  h]. 


48  THE  MEANING  OF  RELATIVITY 


Mass  and  Energy .  An  important  conclusion  can  be 
drawn  from  the  existence  and  significance  of  the  4-vector 
Let  us  imagine  a  body  upon  which  the  electro¬ 
magnetic  field  acts  for  a  time.  In  the  symbolic  figure 
(Fig.  2)  Oxl  designates  the  ^-axis,  and  is  at  the  same 
time  a  substitute  for  the  three  space  axes  Oxv  Ox2,  Oxs  ; 
01  designates  the  real  time  axis.  In  this  diagram  a  body 
of  finite  extent  is  represented,  at  a  definite  time  /,  by  the 
interval  AB  ;  the  whole  space-time  existence  of  the  body 
is  represented  by  a  strip  whose  boundary  is  everywhere 
inclined  less  than  450  to  the  /-axis.  Between  the  time 
sections,  l  =  and  /  =  /2,  but  not  extending  to  them, 
a  portion  of  the  strip  is  shaded.  This  represents  the 
portion  of  the  space-time  manifold  in  which  the  electro¬ 
magnetic  field  acts  upon  the  body,  or  upon  the  electric 
charges  contained  in  it,  the  action  upon  them  being 
transmitted  to  the  body.  We  shall  now  consider  the 
changes  which  take  place  in  the  momentum  and  energy 
of  the  body  as  a  result  of  this  action. 

We  shall  assume  that  the  principles  of  momentum 
and  energy  are  valid  for  the  body.  The  change  in 
momentum,  A  IX)  A  ly,  A  A,  and  the  change  in  energy,  A  E, 
are  then  given  by  the  expressions 


‘1 

M,  -  UW  Xdxdydz  —  Jr^K1dx1dx2dx3dx4 
h 


A  E  = 


/o 


~K \dxyix2dxzdx i 


SPECIAL  RELATIVITY 


49 


Since  the  four-dimensional  element  of  volume  is  an 
invariant,  and  (Kv  K2,  K3,  W4)  forms  a  4-vector,  the  four- 
dimensional  integral  extended  over  the  shaded  portion 
transforms  as  a  4-vector,  as  does  also  the  integral  between 
the  limits  /x  and  /2,  because  the  portion  of  the  region  which 
is  not  shaded  contributes  nothing  to  the  integral.  It 
follows,  therefore,  that  A/*,  A/v,  A I z,  i^E  form  a  4-vector. 
Since  the  quantities  themselves  transform  in  the  same 
way  as  their  increments,  it  follows  that  the  aggregate  of 
the  four  quantities 

A,  4  4  i E 

has  itself  the  properties  of  a  vector;  these  quantities  are 
referred  to  an  instantaneous  condition  of  the  body  (e.g.  at 
the  time  l  =  4. 

This  4-vector  may  also  be  expressed  in  terms  of  the 
mass  ;//,  and  the  velocity  of  the  body,  considered  as  a 
material  particle.  To  form  this  expression,  we  note  first, 
that 

-  ds°-  =  dr2  =  -  (dx2  +  dxd  +  dx2)  -  dx2  =  dl2{\  -  q2')  (38; 

is  an  invariant  which  refers  to  an  infinitely  short  portion 
of  the  four-dimensional  line  which  represents  the  motion 
of  the  material  particle.  The  physical  significance  of  the 
invariant  dr  may  easily  be  given.  If  the  time  axis  is 
chosen  in  such  a  way  that  it  has  the  direction  of  the  line 
differential  which  we  are  considering,  or,  in  other  words, 
if  we  reduce  the  material  particle  to  rest,  we  shall  then 
have  dr  =  dl ;  this  will  therefore  be  measured  by  the 
light-seconds  clock  which  is  at  the  same  place,  and  at 
rest  relatively  to  the  material  particle.  We  therefore  call 
4 


50  THE  MEANING  OF  RELATIVITY 


r  the  proper  time  of  the  material  particle.  As  opposed 
to  dl ,  dr  is  therefore  an  invariant,  and  is  practically 
equivalent  to  dl  for  motions  whose  velocity  is  small 
compared  to  that  of  light.  Hence  we  see  that 

•  •  •  (39) 

dr 

has,  just  as  the  dxvi  the  character  of  a  vector  ;  we  shall 
designate  (uv)  as  the  four-dimensional  vector  (in  brief, 
4-vector)  of  velocity.  Its  components  satisfy,  by  (38), 
the  condition 

=  - 1.  .  .  .  (40) 

We  see  that  this  4-vector,  whose  components  in  the 
ordinary  notation  are 


Qx  Q  y  Q  s  i 

T^l1’  Vi  -  7  7^1-  7^7 


(41) 


is  the  only  4-vector  which  can  be  formed  from  the  velocity 
components  of  the  material  particle  which  are  defined  in 
three  dimensions  by 

_  dx  _  dy  _  dz 

q*  ~  Jl'  ^  tl'  q’~  ic 


We  therefore  see  that 


(42) 


must  be  that  4-vector  which  is  to  be  equated  to  the 
4-vector  of  momentum  and  energy  whose  existence  we 
have  proved  above.  By  equating  the  components,  we 
obtain,  in  three-dimensional  notation, 


SPECIAL  RELATIVITY 


51 


L  = 


x/1  "  f 


E  = 


in 


n/I  -  ?2J 


(43) 


We  recognize,  in  fact,  that  these  components  of 
momentum  agree  with  those  of  classical  mechanics  for 
velocities  which  are  small  compared  to  that  of  light.  For 
large  velocities  the  momentum  increases  more  rapidly 
than  linearly  with  the  velocity,  so  as  to  become  infinite 
on  approaching  the  velocity  of  light. 

If  we  apply  the  last  of  equations  (43)  to  a  material 
particle  at  rest  (q  =  o),  we  see  that  the  energy,  if0,  of  a 
body  at  rest  is  equal  to  its  mass.  Had  we  chosen  the 
second  as  our  unit  of  time,  we  would  have  obtained 


E0  =  me1  .  .  .  (44) 

Mass  and  energy  are  therefore  essentially  alike  ;  they  are 
only  different  expressions  for  the  same  thing.  The  mass 
of  a  body  is  not  a  constant ;  it  varies  with  changes  in  its 
energy.*  We  see  from  the  last  of  equations  (43)  that  E 
becomes  infinite  when  q  approaches  I,  the  velocity  of 
light.  If  we  develop  E  in  powers  of  q 2,  we  obtain, 


r*  in  o  3  4  /  v 

E  =  m  +  —q*  +  | mq*  +  .  .  .  .  (45) 

2  o 

*  The  emission  of  energy  in  radioactive  processes  is  evidently  connected 
with  the  fact  that  the  atomic  weights  are  not  integers.  Attempts  have 
been  made  to  draw  conclusions  from  this  concerning  the  structure  and 
stability  of  the  atomic  nuclei. 


52  THE  MEANING  OF  RELATIVITY 


The  second  term  of  this  expansion  corresponds  to  the 
kinetic  energy  of  the  material  particle  in  classical 
mechanics. 

Equations  of  Motion  of  Material  Particles.  From  (43) 
we  obtain,  by  differentiating  by  the  time  /,  and  using 
the  principle  of  momentum,  in  the  notation  of  three- 
dimensional  vectors, 


This  equation,  which  was  previously  employed  by 
H.  A.  Lorentz  for  the  motion  of  electrons,  has  been 
proved  to  be  true,  with  great  accuracy,  by  experiments 
with  /5-rays. 

Energy  Tensor  of  the  Electromagnetic  Field.  Before  the 
development  of  the  theory  of  relativity  it  was  known 
that  the  principles  of  energy  and  momentum  could 
be  expressed  in  a  differential  form  for  the  electro¬ 
magnetic  field.  The  four-dimensional  formulation  of 
these  principles  leads  to  an  important  conception,  that  of 
the  energy  tensor,  which  is  important  for  the  further 
development  of  the  theory  of  relativity. 

If  in  the  expression  for  the  4-vector  of  force  per  unit 
volume, 


using  the  field  equations  (32),  we  express  in  terms  of 
the  field  intensities,  we  obtain,  after  some  trans¬ 
formations  and  repeated  application  of  the  field  equations 
(32)  and  (33),  the  expression 


(47) 


SPECIAL  RELATIVITY 


53 


where  we  have  written  * 


(48) 


The  physical  meaning  of  equation  (47)  becomes  evident 
if  in  place  of  this  equation  we  write,  using  a  new 
notation, 


iX  = 


'tip  XX : 

'tiP xy 

tipxx 

a(i^) 

tix 

• 

1 

• 

• 

1 

tiz 

m 

• 

Xis*) 

•  • 

2(1%) 

2(>s.) 

X  -  v ) 

dx 

ti)y 

tiz 

2(10 

(47a) 


or,  on  eliminating  the  imaginary, 


K 


x  = 


tip  XX 

tip  xy 

tip  XX 

2£, 

tix 

tiz 

ti/ 

y 


•  •  •  • 


tiSx 

tiSy 

tisx 

tirj 

tix 

tiz 

til 

(47b) 


When  expressed  in  the  latter  form,  we  see  that  the 
first  three  equations  state  the  principle  of  momentum  ; 
Pxx  •  •  •  pxx  are  the  Maxwell  stresses  in  the  electro¬ 
magnetic  field,  and  (bx,  byi  bz)  is  the  vector  momentum 
per  unit  volume  of  the  field.  The  last  of  equations  (47b) 
expresses  the  energy  principle;  s  is  the  vector  flow  of 
energy,  and  77  the  energy  per  unit  volume  of  the  field. 
In  fact,  we  get  from  (48)  by  introducing  the  well-known 
expressions  for  the  components  of  the  field  intensity  from 
electrodynamics, 


*  To  be  summed  for  the  indices  a  and  /3, 


54  THE  MEANING  OF  RELATIVITY 


pxx  —  —  +  -J(h*2  +  h/  +  hy) 

-  exey  +  ie,2  +  e/  +  e,2) 


1 


P xy  ^hj/  P  xz 

—  QxGy 


-  hxhz 


(48  a) 


bx  sx  ©jhs  62  hy 


We  conclude  from  (48)  that  the  energy  tensor  of  the 
electromagnetic  field  is  symmetrical ;  with  this  is  con¬ 
nected  the  fact  that  the  momentum  per  unit  volume  and 
the  flow  of  energy  are  equal  to  each  other  (relation 
between  energy  and  inertia). 

We  therefore  conclude  from  these  considerations  that 
the  energy  per  unit  volume  has  the  character  of  a  tensor. 
This  has  been  proved  directly  only  for  an  electromagnetic 
field,  although  we  may  claim  universal  validity  for  it. 
Maxwell’s  equations  determine  the  electromagnetic  field 
when  the  distribution  of  electric  charges  and  currents  is 
known.  But  we  do  not  know  the  laws  which  govern 
the  currents  and  charges.  We  do  know,  indeed,  that 
electricity  consists  of  elementary  particles  (electrons, 
positive  nuclei),  but  from  a  theoretical  point  of  view  we 
cannot  comprehend  this.  We  do  not  know  the  energy 
factors  which  determine  the  distribution  of  electricity  in 
particles  of  definite  size  and  charge,  and  all  attempts  to 
complete  the  theory  in  this  direction  have  failed.  If  then 
we  can  build  upon  Maxwell’s  equations  in  general,  the 


SPECIAL  RELATIVITY 


55 


energy  tensor  of  the  electromagnetic  field  is  known  only 
outside  the  charged  particles.*  In  these  regions,  outside 
of  charged  particles,  the  only  regions  in  which  we  can 
believe  that  we  have  the  complete  expression  for  the 
energy  tensor,  we  have,  by  (47), 


IT, 


txv 


=  O 


(47c) 


General  Expressions  for  the  Conservation  Principles.  We 
can  hardly  avoid  making  the  assumption  that  in  all  other 
cases,  also,  the  space  distribution  of  energy  is  given  by  a 
symmetrical  tensor,  T)X ,  and  that  this  complete  energy 
tensor  everywhere  satisfies  the  relation  (47c).  At  any 
rate  we  shall  see  that  by  means  of  this  assumption  we 
obtain  the  correct  expression  for  the  integral  energy 
principle. 

Let  us  consider  a  spatially  bounded,  closed  system, 
which,  four-dimensionally,  we  may  represent  as  a  strip, 
outside  of  which  the  T^v  vanish.  Integrate  equation 
(47c)  over  a  space  section.  Since  the  integrals  of 


tT. 


^1 


r — ,  -r— ^ -  and  -r— 1 —  vanish  because  the  Tuv  vanish  at  the 
oxl  ’  ^x»  Lr3  ^ 


limits  of  integration,  we  obtain 


(I 


57I  I  T^dxydx^ 


o 


(49) 


Inside  the  parentheses  are  the  expressions  for  the 


*  It  has  been  attempted  to  remedy  this  lack  of  knowledge  by  considering 
the  charged  particles  as  proper  singularities.  But  in  my  opinion  this  means 
giving  up  a  real  understanding  of  the  structure  of  matter.  It  seems  to  me 
much  better  to  give  in  to  our  present  inability  rather  than  to  be  satisfied 
by  a  solution  that  is  only  apparent. 


56  THE  MEANING  OF  RELATIVITY 

momentum  of  the  whole  system,  multiplied  by  z,  together 
with  the  negative  energy  of  the  system,  so  that  (49) 
expresses  the  conservation  principles  in  their  integral 
form.  That  this  gives  the  right  conception  of  energy  and 

I 


the  conservation  principles  will  be  seen  from  the  following 
considerations. 

Phenomenological  Representation  of  the 
Energy  Tensor  of  Matter. 

H ydrodynamical  Equations.  We  know  that  matter  is 
built  up  of  electrically  charged  particles,  but  we  do  not 


SPECIAL  RELATIVITY 


57 


know  the  laws  which  govern  the  constitution  of  these 
particles.  In  treating  mechanical  problems,  we  are  there¬ 
fore  obliged  to  make  use  of  an  inexact  description  of 
matter,  which  corresponds  to  that  of  classical  mechanics. 
The  density  <7,  of  a  material  substance  and  the  hydro- 
dynamical  pressures  are  the  fundamental  concepts  upon 
which  such  a  description  is  based. 

Let  cr0  be  the  density  of  matter  at  a  place,  estimated 
with  reference  to  a  system  of  co-ordinates  moving  with 
the  matter.  Then  cr0,  the  density  at  rest,  is  an  invariant. 
If  we  think  of  the  matter  in  arbitrary  motion  and  neglect 
the  pressures  (particles  of  dust  in  vacuo ,  neglecting  the 
size  of  the  particles  and  the  temperature),  then  the  energy 
tensor  will  depend  only  upon  the  velocity  components, 
uv  and  cr0.  We  secure  the  tensor  character  of  T^v  by 
putting 

T,V  &Q  M^jUy  .  .  .  (50) 

in  which  the  u in  the  three-dimensional  representation, 
are  given  by  (41).  In  fact,  it  follows  from  (50)  that  for 
q  —  o,  7"44  =  -  <70  (equal  to  the  negative  energy  per  unit 
volume),  as  it  should,  according  to  the  theorem  of  the 
equivalence  of  mass  and  energy,  and  according  to  the 
physical  interpretation  of  the  energy  tensor  given  above. 
If  an  external  force  (four-dimensional  vector,  WJ  acts 
upon  the  matter,  by  the  principles  of  momentum  and 
energy  the  equation 


58  THE  MEANING  OF  RELATIVITY 


must  hold.  We  shall  now  show  that  this  equation  leads 
to  the  same  law  of  motion  of  a  material  particle  as  that 
already  obtained.  Let  us  imagine  the  matter  to  be  of 
infinitely  small  extent  in  space,  that  is,  a  four-dimensional 
thread  ;  then  by  integration  over  the  whole  thread  with 
respect  to  the  space  co-ordinates  a q,  x2,  ;r3,  we  obtain 


^K‘1dx1dx2dx2 


^dL*dx,  dx.dxo  = 

J  Ltq  1  ^  3 


.d 
1 — 

dl 


dx,  dx.,  ,  j 
dr  dr  123 


Now  j dxYdxtflx%dx±  is  an  invariant,  as  is,  therefore,  also 

r • 

j (T^dx^dx^dx%dx^.  We  shall  calculate  this  integral,  first 

with  respect  to  the  inertial  system  which  we  have  chosen, 
and  second,  with  respect  to  a  system  relatively  to  which 
the  matter  has  the  velocity  zero.  The  integration  is  to 
be  extended  over  a  filament  of  the  thread  for  which  cr0 
may  be  regarded  as  constant  over  the  whole  section.  If 
the  space  volumes  of  the  filament  referred  to  the  two 
systems  are  dV  and  dV0  respectively,  then  we  have 


L0dV<il  = 


[cr^dVfjdT 


and  therefore  also 


Jcr 0dV  =  =  jd/n  i 


.  dr 

dx \ 


If  we  substitute  the  right-hand  side  for  the  left-hand 

,  .  dx 

side  in  the  former  integral,  and  put  ~  outside  the  sign 

dr 


SPECIAL  RELATIVITY 


59 


of  integration,  we  obtain, 

is  _  d(,Jx i\  _  d  (  m \ 

K *  ~  diK^)  ~  dkjnr?) 

We  see,  therefore,  that  the  generalized  conception  of  the 
energy  tensor  is  in  agreement  with  our  former  result. 

The  Eulerian  Equations  for  Perfect  Fluids.  In  order 
to  get  nearer  to  the  behaviour  of  real  matter  we  must  add 
to  the  energy  tensor  a  term  which  corresponds  to  the 
pressures.  The  simplest  case  is  that  of  a  perfect  fluid  in 
which  the  pressure  is  determined  by  a  scalar  p.  Since 
the  tangential  stresses  pxy,  etc.,  vanish  in  this  case,  the 
contribution  to  the  energy  tensor  must  be  of  the  form 
p8vll.  We  must  therefore  put 

T ^  —  <?upuv  +  pS^  .  .  (5  0 

At  rest,  the  density  of  the  matter,  or  the  energy  per  unit 
volume,  is  in  this  case,  not  a  but  a  -  p.  For 


( 7 


dx 4  dx  i 
dr  dr 


a  -  p. 


In  the  absence  of  any  force,  we  have 


dT, 


fXV 


dx„ 


duu  d(auh) 
<ruvZ — -  +  u)L  + 


¥  


dx, 


dXv 


dx, 


=  o. 


If  we  multiply  this  equation  by  ua 
the  fs  we  obtain,  using  (40), 


and  sum  for 


60  THE  MEANING  OF  RELATIVITY 


where  we  have  put  This  is  the  equation  of 


'bxfL  dr 


dr 


continuity,  which  differs  from  that  of  classical  mechanics 

by  the  term  which,  practically,  is  vanishingly  small. 
dr 

Observing  (52),  the  conservation  principles  take  the  form 


+  «, 


dp 

Ldr 


+ 


The  equations  for  the  first  three  indices  evidently  corre¬ 
spond  to  the  Eulerian  equations.  That  the  equations 
(52)  and  (53)  correspond,  to  a  first  approximation,  to  the 
hydrodynamical  equations  of  classical  mechanics,  is  a 
further  confirmation  of  the  generalized  energy  principle. 
The  density  of  matter  and  of  energy  has  the  character  of 
a  symmetrical  tensor. 


LECTURE  III 


THE  GENERAL  THEORY  OF  RELATIVITY 

ALL  of  the  previous  considerations  have  been  based 
upon  the  assumption  that  all  inertial  systems  are 
equivalent  for  the  description  of  physical  phenomena,  but 
that  they  are  preferred,  for  the  formulation  of  the  laws 
of  nature,  to  spaces  of  reference  in  a  different  state  of 
motion.  We  can  think  of  no  cause  for  this  preference 
for  definite  states  of  motion  to  all  others,  according  to 
our  previous  considerations,  either  in  the  perceptible 
bodies  or  in  the  concept  of  motion  ;  on  the  contrary,  it 
must  be  regarded  as  an  independent  property  of  the 
space-time  continuum.  The  principle  of  inertia,  in 
particular,  seems  to  compel  us  to  ascribe  physically 
objective  properties  to  the  space-time  continuum.  Just 
as  it  was  necessary  from  the  Newtonian  standpoint  to 
make  both  the  statements,  tempus  est  absolutum ,  spatium 
est  absolutum ,  so  from  the  standpoint  of  the  special  theory 
of  relativity  we  must  say,  continuum  spatii  et  temporis  est 
absolutum.  In  this  latter  statement  absolutum  means  not 
only  “physically  real,”  but  also  “independent  in  its 
physical  properties,  having  a  physical  effect,  but  not  itself 
influenced  by  physical  conditions.” 

As  long  as  the  principle  of  inertia  is  regarded  as  the 

61 


62  THE  MEANING  OF  RELATIVITY 


keystone  of  physics,  this  standpoint  is  certainly  the  only 
one  which  is  justified.  But  there  are  two  serious  criticisms 
of  the  ordinary  conception.  In  the  first  place,  it  is  contrary 
to  the  mode  of  thinking  in  science  to  conceive  of  a  thing 
(the  space-time  continuum)  which  acts  itself,  but  which 
cannot  be  acted  upon.  This  is  the  reason  why  E.  Mach 
was  led  to  make  the  attempt  to  eliminate  space  as  an 
active  cause  in  the  system  of  mechanics.  According  to 
him,  a  material  particle  does  not  move  in  unaccelerated 
motion  relatively  to  space,  but  relatively  to  the  centre  of 
all  the  other  masses  in  the  universe;  in  this  way  the 
series  of  causes  of  mechanical  phenomena  was  closed,  in 
contrast  to  the  mechanics  of  Newton  and  Galileo.  In 
order  to  develop  this  idea  within  the  limits  of  the  modern 
theory  of  action  through  a  medium,  the  properties  of 
the  space-time  continuum  which  determine  inertia  must 
be  regarded  as  field  properties  of  space,  analogous  to 
the  electromagnetic  field.  The  concepts  of  classical 
mechanics  afford  no  way  of  expressing  this.  For  this 
reason  Mach’s  attempt  at  a  solution  failed  for  the  time 
being.  We  shall  come  back  to  this  point  of  view  later. 
In  the  second  place,  classical  mechanics  indicates  a 
limitation  which  directly  demands  an  extension  of  the 
principle  of  relativity  to  spaces  of  reference  which  are  not 
in  uniform  motion  relatively  to  each  other.  The  ratio  of 
the  masses  of  two  bodies  is  defined  in  mechanics  in  two 
ways  which  differ  from  each  other  fundamentally ;  in  the 
first  place,  as  the  reciprocal  ratio  of  the  accelerations 
which  the  same  motional  force  imparts  to  them  (inert 
mass),  and  in  the  second  place,  as  the  ratio  of  the  forces 


THE  GENERAL  THEORY 


63 


which  act  upon  them  in  the  same  gravitational  held 
(gravitational  mass).  The  equality  of  these  two  masses, 
so  differently  defined,  is  a  fact  which  is  confirmed  by 
experiments  of  very  high  accuracy  (experiments  of  Edtvos), 
and  classical  mechanics  offers  no  explanation  for  this 
equality.  It  is,  however,  clear  that  science  is  fully  justified 
in  assigning  such  a  numerical  equality  only  after  this 
numerical  equality  is  reduced  to  an  equality  of  the  real 
nature  of  the  two  concepts. 

That  this  object  may  actually  be  attained  by  an  exten¬ 
sion  of  the  principle  of  relativity,  follows  from  the  follow¬ 
ing  consideration.  A  little  reflection  will  show  that  the 
theorem  of  the  equality  of  the  inert  and  the  gravitational 
mass  is  equivalent  to  the  theorem  that  the  acceleration 
imparted  to  a  body  by  a  gravitational  field  is  independent 
of  the  nature  of  the  body.  For  Newton’s  equation  of 
motion  in  a  gravitational  field,  written  out  in  full,  is 

(Inert  mass).  (Acceleration)  =  (Intensity  of  the 

gravitational  field)  .  (Gravitational  mass). 

It  is  only  when  there  is  numerical  equality  between  the 
inert  and  gravitational  mass  that  the  acceleration  is  in¬ 
dependent  of  the  nature  of  the  body.  Let  now  K  be  an 
inertial  system.  Masses  which  are  sufficiently  far  from 
each  other  and  from  other  bodies  are  then,  with  respect 
to  Y,  free  from  acceleration.  We  shall  also  refer  these 
masses  to  a  system  of  co-ordinates  K\  uniformly  acceler¬ 
ated  with  respect  to  K.  Relatively  to  K'  all  the  masses 
have  equal  and  parallel  accelerations  ;  with  respect  to  K' 
they  behave  just  as  if  a  gravitational  field  were  present  and 


64  THE  MEANING  OF  RELATIVITY 


K'  were  unaccelerated.  Overlooking  for  the  present  the 
question  as  to  the  “  cause  ”  of  such  a  gravitational  field, 
which  will  occupy  us  later,  there  is  nothing  to  prevent 
our  conceiving  this  gravitational  field  as  real,  that  is,  the 
conception  that  K'  is  “  at  rest  ”  and  a  gravitational  field 
is  present  we  may  consider  as  equivalent  to  the  concep¬ 
tion  that  only  K  is  an  “  allowable  ”  system  of  co-ordinates 
and  no  gravitational  field  is  present.  The  assumption  of 
the  complete  physical  equivalence  of  the  systems  of  co¬ 
ordinates,  K  and  K\  we  call  the  “  principle  of  equival¬ 
ence;”  this  principle  is  evidently  intimately  connected 
with  the  theorem  of  the  equality  between  the  inert  and 
the  gravitational  mass,  and  signifies  an  extension  of  the 
principle  of  relativity  to  co-ordinate  systems  which  are 
in  non-uniform  motion  relatively  to  each  other.  In  fact, 
through  this  conception  we  arrive  at  the  unity  of  the 
nature  of  inertia  and  gravitation.  For  according  to  our 
way  of  looking  at  it,  the  same  masses  may  appear  to  be 
either  under  the  action  of  inertia  alone  (with  respect  to 
K)  or  under  the  combined  action  of  inertia  and  gravita¬ 
tion  (with  respect  to  K).  The  possibility  of  explaining 
the  numerical  equality  of  inertia  and  gravitation  by  the 
unity  of  their  nature  gives  to  the  general  theory  of 
relativity,  according  to  my  conviction,  such  a  superiority 
over  the  conceptions  of  classical  mechanics,  that  all  the 
difficulties  encountered  in  development  must  be  considered 
as  small  in  comparison. 

What  justifies  us  in  dispensing  with  the  preference 
for  inertial  systems  over  all  other  co-ordinate  systems,  a 
preference  that  seems  so  securely  established  by  experi- 


THE  GENERAL  THEORY  65 

ment  based  upon  the  principle  of  inertia  ?  The  weakness 
of  the  principle  of  inertia  lies  in  this,  that  it  involves  an 
argument  in  a  circle  :  a  mass  moves  without  acceleration 
if  it  is  sufficiently  far  from  other  bodies;  we  know  that 
it  is  sufficiently  far  from  other  bodies  only  by  the  fact 
that  it  moves  without  acceleratioa  Are  there,  in  general, 
any  inertial  systems  for  very  extended  portions  of  the 
space-time  continuum,  or,  indeed,  for  the  whole  universe? 
We  may  look  upon  the  principle  of  inertia  as  established, 
to  a  high  degree  of  approximation,  for  the  space  of  our 
planetary  system,  provided  that  we  neglect  the  perturba¬ 
tions  due  to  the  sun  and  planets.  Stated  more  exactly, 
there  are  finite  regions,  where,  with  respect  to  a  suitably 
chosen  space  of  reference,  material  particles  move  freely 
without  acceleration,  and  in  which  the  laws  of  the  special 
theory  of  relativity,  which  have  been  developed  above, 
hold  with  remarkable  accuracy.  Such  regions  we  shall 
call  “Galilean  regions.”  We  shall  proceed  from  the 
consideration  of  such  regions  as  a  special  case  of  known 
properties. 

The  principle  of  equivalence  demands  that  in  dealing 
with  Galilean  regions  we  may  equally  well  make  use  of 
non-inertial  systems,  that  is,  such  co-ordinate  systems  as, 
relatively  to  inertial  systems,  are  not  free  from  accelera¬ 
tion  and  rotation.  If,  further,  we  are  going  to  do  away 
completely  with  the  difficult  question  as  to  the  objective 
reason  for  the  preference  of  certain  systems  of  co-ordinates, 
then  we  must  allow  the  use  of  arbitrarily  moving  systems 
of  co-ordinates.  As  soon  as  we  make  this  attempt  seriously 


5 


66  THE  MEANING  OF  RELATIVITY 


we  come  into  conflict  with  that  physical  interpretation  of 
space  and  time  to  which  we  were  led  by  the  special  theory 
of  relativity.  For  let  K'  be  a  system  of  co-ordinates  whose 
/-axis  coincides  with  the  -S'-axis  of  K ,  and  which  rotates 
about  the  latter  axis  with  constant  angular  velocity.  Are 
the  configurations  of  rigid  bodies,  at  rest  relatively  to  K\ 
in  accordance  with  the  laws  of  Euclidean  geometry? 
Since  K'  is  not  an  inertial  system,  we  do  not  know 
directly  the  laws  of  configuration  of  rigid  bodies  with 
respect  to  K',  nor  the  laws  of  nature,  in  general.  But 
we  do  know  these  laws  with  respect  to  the  inertial  system 
K:  and  we  can  therefore  estimate  them  with  respect  to  K'. 
Imagine  a  circle  drawn  about  the  origin  in  the  x'y  plane 
of  K\  and  a  diameter  of  this  circle.  Imagine,  further,  that 
we  have  given  a  large  number  of  rigid  rods,  all  equal  to 
each  other.  We  suppose  these  laid  in  series  along  the 
periphery  and  the  diameter  of  the  circle,  at  rest  relatively 
to  K' .  If  U  is  the  number  of  these  rods  along  the  peri¬ 
phery,  D  the  number  along  the  diameter,  then,  if  K  does 
not  rotate  relatively  to  K,  we  shall  have 

U 

d  ~ 77  • 

But  if  K  rotates  we  get  a  different  result.  Suppose 
that  at  a  definite  time  t}  of  K  we  determine  the  ends  of 
all  the  rods.  With  respect  to  K  all  the  rods  upon  the 
periphery  experience  the  Lorentz  contraction,  but  the 
rods  upon  the  diameter  do  not  experience  this  contrac- 


THE  GENERAL  THEORY 


67 


tion  (along  their  lengths  !).*  It  therefore  follows  that 

U 

D>lr ■ 

It  therefore  follows  that  the  laws  of  configuration  of 
rigid  bodies  with  respect  to  K'  do  not  agree  with  the 
laws  of  configuration  of  rigid  bodies  that  are  in  accord¬ 
ance  with  Euclidean  geometry.  If,  further,  we  place  two 
similar  clocks  (rotating  withTT),  one  upon  the  periphery, 
and  the  other  at  the  centre  of  the  circle,  then,  judged 
from  Ky  the  clock  on  the  periphery  will  go  slower  than 
the  clock  at  the  centre.  The  same  thing  must  take  place, 
judged  from  K\  if  we  define  time  with  respect  to  K'  in 
a  not  wholly  unnatural  way,  that  is,  in  such  a  way  that 
the  laws  with  respect  to  K'  depend  explicitly  upon  the 
time.  Space  and  time,  therefore,  cannot  be  defined 
with  respect  to  K'  as  they  were  in  the  special  theory  of 
relativity  with  respect  to  inertial  systems.  But,  accord¬ 
ing  to  the  principle  of  equivalence,  K'  is  also  to  be  con¬ 
sidered  as  a  system  at  rest,  with  respect  to  which  there 
is  a  gravitational  field  (field  of  centrifugal  force,  and 
force  of  Coriolis).  We  therefore  arrive  at  the  result : 
the  gravitational  field  influences  and  even  determines  the 
metrical  laws  of  the  space-time  continuum.  If  the  laws 
of  configuration  of  ideal  rigid  bodies  are  to  be  expressed 
geometrically,  then  in  the  presence  of  a  gravitational 
field  the  geometry  is  not  Euclidean. 

*  These  considerations  assume  that  the  behaviour  of  rods  and  clocks 
depends  only  upon  velocities,  and  not  upon  accelerations,  or,  at  least,  that 
the  influence  of  acceleration  does  not  counteract  that  of  velocity. 


68  THE  MEANING  OF  RELATIVITY 


The  case  that  we  have  been  considering  is  analogous 
to  that  which  is  presented  in  the  two-dimensional  treat¬ 
ment  of  surfaces.  It  is  impossible  in  the  latter  case 
also,  to  introduce  co-ordinates  on  a  surface  (e.g.  the 
surface  of  an  ellipsoid)  which  have  a  simple  metrical 
significance,  while  on  a  plane  the  Cartesian  co-ordinates, 
xv  x2,  signify  directly  lengths  measured  by  a  unit 
measuring  rod.  Gauss  overcame  this  difficulty,  in  his 
theory  of  surfaces,  by  introducing  curvilinear  co-ordinates 
which,  apart  from  satisfying  conditions  of  continuity, 
were  wholly  arbitrary,  and  afterwards  these  co-ordinates 
were  related  to  the  metrical  properties  of  the  surface. 
In  an  analogous  way  we  shall  introduce  in  the  general 
theory  of  relativity  arbitrary  co-ordinates,  xv  x2,  xv  x^ 
which  shall  number  uniquely  the  space-time  points,  so 
that  neighbouring  events  are  associated  with  neighbour¬ 
ing  values  of  the  co-ordinates  ;  otherwise,  the  choice  of 
co-ordinates  is  arbitrary.  We  shall  be  true  to  the 
principle  of  relativity  in  its  broadest  sense  if  we  give 
such  a  form  to  the  laws  that  they  are  valid  in  every 
such  four-dimensional  system  of  co-ordinates,  that  is,  if 
the  equations  expressing  the  laws  are  co-variant  with 
respect  to  arbitrary  transformations. 

The  most  important  point  of  contact  between  Gauss’s 
theory  of  surfaces  and  the  general  theory  of  relativity 
lies  in  the  metrical  properties  upon  which  the  concepts 
of  both  theories,  in  the  main,  are  based.  In  the  case 
of  the  theory  of  surfaces,  Gauss’s  argument  is  as  follows. 
Plane  geometry  may  be  based  upon  the  concept  of  the 
distance  ds,  between  two  indefinitely  near  points.  The 


THE  GENERAL  THEORY 


69 


concept  of  this  distance  is  physically  significant  because 
the  distance  can  be  measured  directly  by  means  of  a 
rigid  measuring  rod.  By  a  suitable  choice  of  Cartesian 
co-ordinates  this  distance  may  be  expressed  by  the 
formula  ds2  =  dx 2  +  dx 22.  We  may  base  upon  this 
quantity  the  concepts  of  the  straight  line  as  the  geodesic 
(h\ds  =  o),  the  interval,  the  circle,  and  the  angle,  upon 
which  the  Euclidean  plane  geometry  is  built.  A 
geometry  may  be  developed  upon  another  continuously 
curved  surface,  if  we  observe  that  an  infinitesimally 
small  portion  of  the  surface  may  be  regarded  as  plane, 
to  within  relatively  infinitesimal  quantities.  There  are 
Cartesian  co-ordinates,  Xlt  X%t  upon  such  a  small 
portion  of  the  surface,  and  the  distance  between  two 
points,  measured  by  a  measuring  rod,  is  given  by 

ds1  =  dX ,2  +  dX*. 

If  we  introduce  arbitrary  curvilinear  co-ordinates,  xY,  x2 , 
on  the  surface,  then  dXlt  dX2 ,  may  be  expressed  linearly 
in  terms  of  dxlt  dx2.  Then  everywhere  upon  the  sur¬ 
face  we  have 

ds2  =  gndx^  +  2gudx1dx2  +  g^dxg 

where  gn,  g12,  g22  are  determined  by  the  nature  of  the 
surface  and  the  choice  of  co-ordinates  ;  if  these  quantities 
are  known,  then  it  is  also  known  how  networks  of  rigid 
rods  may  be  laid  upon  the  surface.  In  other  words,  the 
geometry  of  surfaces  may  be  based  upon  this  expression 
for  ds 2  exactly  as  plane  geometry  is  based  upon  the 
corresponding  expression. 

There  are  analogous  relations  in  the  four-dimensional 


70  THE  MEANING  OF  RELATIVITY 


space-time  continuum  of  physics.  In  the  immediate 
neighbourhood  of  an  observer,  falling  freely  in  a  gravi¬ 
tational  field,  there  exists  no  gravitational  field.  We 
can  therefore  always  regard  an  infinitesimally  small 
region  of  the  space-time  continuum  as  Galilean.  For 
such  an  infinitely  small  region  there  will  be  an  inertial 
system  (with  the  space  co-ordinates,  Xlt  X2,  AG,  and  the 
time  co-ordinate  A”4)  relatively  to  which  we  are  to  regard 
the  laws  of  the  special  theory  of  relativity  as  valid.  The 
quantity  which  is  directly  measurable  by  our  unit 
measuring  rods  and  clocks, 

dx2  +  dX A  +  dX 32  -  dX2 
or  its  negative, 

ds1  =  -  dX2  -  dX2  -  dX2  +  dX 2  .  (54) 

is  therefore  a  uniquely  determinate  invariant  for  two 
neighbouring  events  (points  in  the  four-dimensional 
continuum),  provided  that  we  use  measuring  rods  that 
are  equal  to  each  other  when  brought  together  and 
superimposed,  and  clocks  whose  rates  are  the  same 
when  they  are  brought  together.  In  this  the  physical 
assumption  is  essential  that  the  relative  lengths  of  two 
measuring  rods  and  the  relative  rates  of  two  clocks  are 
independent,  in  principle,  of  their  previous  history.  But 
this  assumption  is  certainly  warranted  by  experience; 
if  it  did  not  hold  there  could  be  no  sharp  spectral  lines  ; 
for  the  single  atoms  of  the  same  element  certainly  do 
not  have  the  same  history,  and  it  would  be  absurd  to 
suppose  any  relative  difference  in  the  structure  of  the 


THE  GENERAL  THEORY 


71 


single  atoms  due  to  their  previous  history  if  the  mass 
and  frequencies  of  the  single  atoms  of  the  same  element 
were  always  the  same. 

Space-time  regions  of  finite  extent  are,  in  general, 
not  Galilean,  so  that  a  gravitational  field  cannot  be  done 
away  with  by  any  choice  of  co-ordinates  in  a  finite 
region.  There  is,  therefore,  no  choice  of  co-ordinates 
for  which  the  metrical  relations  of  the  special  theory  of 
relativity  hold  in  a  finite  region.  But  the  invariant  ds 
always  exists  for  two  neighbouring  points  (events)  of 
the  continuum.  This  invariant  ds  may  be  expressed  in 
arbitrary  co-ordinates.  If  one  observes  that  the  local 
dXv  may  be  expressed  linearly  in  terms  of  the  co¬ 
ordinate  differentials  dx„  ds 2  may  be  expressed  in  the 
form 

ds1  =  g^dx/lXy  .  .  •  (55) 

The  functions  g^v  describe,  with  respect  to  the  arbit¬ 
rarily  chosen  system  of  co-ordinates,  the  metrical  rela¬ 
tions  of  the  space-time  continuum  and  also  the 
gravitational  field.  As  in  the  special  theory  of  relativity, 
we  have  to  discriminate  between  time-like  and  space¬ 
like  line  elements  in  the  four-dimensional  continuum  ; 
owing  to  the  change  of  sign  introduced,  time-like 
line  elements  have  a  real,  space-like  line  elements  an 
imaginary  ds.  The  time-like  ds  can  be  measured  directly 
by  a  suitably  chosen  clock. 

According  to  what  has  been  said,  it  is  evident  that 
the  formulation  of  the  general  theory  of  relativity 
assumes  a  generalization  of  the  theory  of  invariants  and 
the  theory  of  tensors;  the  question  is  raised  as  to  the 


72  THE  MEANING  OF  RELATIVITY 


form  of  the  equations  which  are  co-variant  with  respect 
to  arbitrary  point  transformations.  The  generalized 
calculus  of  tensors  was  developed  by  mathematicians 
long  before  the  theory  of  relativity.  Riemann  first 
extended  Gauss’s  train  of  thought  to  continua  of  any 
number  of  dimensions ;  with  prophetic  vision  he  saw 
the  physical  meaning  of  this  generalization  of  Euclid’s 
geometry.  Then  followed  the  development  of  the  theory 
in  the  form  of  the  calculus  of  tensors,  particularly  by 
Ricci  and  Levi-Civita.  This  is  the  place  for  a  brief 
presentation  of  the  most  important  mathematical  con¬ 
cepts  and  operations  of  this  calculus  of  tensors. 

We  designate  four  quantities,  which  are  defined  as 
functions  of  the  xv  with  respect  to  every  system  of  co¬ 
ordinates,  as  components,  Auy  of  a  contra-variant  vector, 
if  they  transform  in  a  change  of  co-ordinates  as  the  co¬ 
ordinate  differentials  dxv.  We  therefore  have 


(W 

A*'  =  z-^A\ 


(so 


Besides  these  contra-variant  vectors,  there  are  also  co¬ 
variant  vectors.  If  Bv  are  the  components  of  a  co-variant 
vector,  these  vectors  are  transformed  according  to  the 
rule 


B\  = 


(57) 


The  definition  of  a  co-variant  vector  is  chosen  in  such  a 
way  that  a  co-variant  vector  and  a  contra-variant  vector 
together  form  a  scalar  according  to  the  scheme, 


< ft  =  BVAV  (summed  over  the  v). 


THE  GENERAL  THEORY 


73 


Accordingly, 


B\A*'  =  ^ 


~dx 


BA f*  =  BnA \ 


p 


In  particular,  the  derivatives  of  a  scalar  6,  are  com- 

ponents  of  a  co-variant  vector,  which,  with  the  co-ordinate 

differentials,  form  the  scalar  ;  we  see  from  this 

tea 

example  how  natural  is  the  definition  of  the  co-variant 
vectors. 

There  are  here,  also,  tensors  of  any  rank,  which  may 
have  co-variant  or  contra-variant  character  with  respect 
to  each  index  ;  as  with  vectors,  the  character  is  desig¬ 
nated  by  the  position  of  the  index.  For  example,  A / 
denotes  a  tensor  of  the  second  rank,  which  is  co-variant 
with  respect  to  the  index  /i,  and  contra-variant  with  re¬ 
spect  to  the  index  v.  The  tensor  character  indicates 
that  the  equation  of  transformation  is 


^  a' 


(58) 


Tensors  may  be  formed  by  the  addition  and  subtraction 
of  tensors  of  equal  rank  and  like  character,  as  in  the 
theory  of  invariants  of  orthogonal  linear  substitutions,  for 
example, 

a;  +  b;=  q.  .  .  .  (59) 

The  proof  of  the  tensor  character  of  C*  depends  upon  (58). 

Tensors  may  be  formed  by  multiplication,  keeping  the 
character  of  the  indices,  just  as  in  the  theory  of  invariants 
of  linear  orthogonal  transformations,  for  example, 

.  (60) 


rv 

'■''/ACTT* 


74  THE  MEANING  OF  RELATIVITY 


The  proof  follows  directly  from  the  rule  of  transforma¬ 
tion. 

Tensors  may  be  formed  by  contraction  with  respect  to 
two  indices  of  different  character,  for  example, 

A%t  =  BaT.  .  .  .  (6 1 ) 

The  tensor  character  of  A£ar  determines  the  tensor 
character  of  BaT.  Proof — 


_  3*.  3£V 


Lrs  ~dXt 
~dx'a  dx'T 


a 

ast • 


The  properties  of  symmetry  and  skew-symmetry  of  a 
tensor  with  respect  to  two  indices  of  like  character  have 
the  same  significance  as  in  the  theory  of  invariants. 

With  this,  everything  essential  has  been  said  with 
regard  to  the  algebraic  properties  of  tensors. 

The  Fundamental  Tensor.  It  follows  from  the  invari¬ 
ance  of  ds 2  for  an  arbitrary  choice  of  the  dxv,  in  connexion 
with  the  condition  of  symmetry  consistent  with  (55),  that 
the  g^v  are  components  of  a  symmetrical  co-variant  tensor 
(Fundamental  Tensor).  Let  us  form  the  determinant, 
g,  of  the  g^vi  and  also  the  minors,  divided  by  g,  cor¬ 
responding  to  the  single  g^.  These  minors,  divided  by 
g}  will  be  denoted  by  g and  their  co-variant  character 
is  not  yet  known.  Then  we  have 


O' 


<rtf  =  =  1  Ur  a  7  ^ 

a  O  if  a  ft 


(62) 


If  we  form  the  infinitely  small  quantities  (co-variant 
vectors) 

—  g^oF^o.  •  •  •  (^3) 


THE  GENERAL  THEORY  75 


multiply  by  g'x&  and  sum  over  the  //,,  we  obtain,  by  the 
use  of  (62), 

dxi  =  g^d^.  .  .  ■  (64) 

Since  the  ratios  of  the  d^  are  arbitrary,  and  the  dx$  as 
well  as  the  dx^  are  components  of  vectors,  it  follows  that 
the  g*v  are  the  components  of  a  contra-variant  tensor  * 
(contra-variant  fundamental  tensor).  The  tensor  character 
of  Sf  (mixed  fundamental  tensor)  accordingly  follows, 
by  (62).  By  means  of  the  fundamental  tensor,  instead 
of  tensors  with  co-variant  index  character,  we  can 
introduce  tensors  with  contra-variant  index  character, 
and  conversely.  For  example, 

=  g»*Aa 

A  = 

Ta  =  . 

fJ-  5  fj-v 

Volume  Invariants.  The  volume  element 


S  dxYdx.L  dxzdxA  =  dx 

is  not  an  invariant.  For  by  Jacobi’s  theorem, 


dx  = 


dx’* 

dx. 


dx. 


(65) 


& 

*  If  we  multiply  (64)  by  ,  sum  over  the  £,  and  replace  the  dj-n  by  a 
transformation  to  the  accented  system,  we  obtain 


dx'a 


dx'q-  'dx'a 
dx^  dxp 


g^civ 


<T • 


The  statement  made  above  follows  from  this,  since,  by  (64),  we  must  also 
have  dx'a  =  g^'d}-' a ,  and  both  equations  must  hold  for  every  choice  of  the 

d£'cr. 


76  THE  WEANING  OF  RELATIVITY 


But  we  can  complement  dx  so  that  it  becomes  an  in¬ 
variant.  If  we  form  the  determinant  of  the  quantities 


,  ^xa  Dx,, 

~  ix'^  7>x'g^ 

we  obtain,  by  a  double  application  of  the  theorem  of 
multiplication  of  determinants, 


O' 

a 


cr 

<b  MV 


We  therefore  get  the  invariant, 


Jgdx  =  Jgdx. 


Formation  of  Te?isors  by  Differentiation.  Although 
the  algebraic  operations  of  tensor  formation  have  proved 
to  be  as  simple  as  in  the  special  case  of  invariance  with 
respect  to  linear  orthogonal  transformations,  nevertheless 
in  the  general  case,  the  invariant  differential  operations 
are,  unfortunately,  considerably  more  complicated.  The 
reason  for  this  is  as  follows.  If  A *  is  a  contra-variant 

vector,  the  coefficients  of  its  transformation,  are  in- 

Dxv 

dependent  of  position  only  if  the  transformation  is  a  linear 

DA* 

one.  For  then  the  vector  components,  A*  +  — — dx a,  at 

oXa 

a  neighbouring  point  transform  in  the  same  way  as  the 
A*,  from  which  follows  the  vector  character  of  the  vector 


differentials,  and  the  tensor  character  of 


DA* 


Dxf 

Dxv 


are  variable  this  is  no  longer  true. 


But  if  the 


THE  GENERAL  THEORY 


77 


That  there  are,  nevertheless,  in  the  general  case,  in¬ 
variant  differential  operations  for  tensors,  is  recognized 
most  satisfactorily  in  the  following  way,  introduced  by 
Levi-Civita  and  Weyl.  Let  (A*)  be  a  contra -variant  vector 
whose  components  are  given  with  respect  to  the  co¬ 
ordinate  system  of  the  xv.  Let  P1  and  P2  be  two  in¬ 
finitesimally  near  points  of  the  continuum.  For  the 
infinitesimal  region  surrounding  the  point  Pv  there  is, 
according  to  our  way  of  considering  the  matter,  a  co¬ 
ordinate  system  of  the  Xv  (with  imaginary  ^-co¬ 
ordinates)  for  which  the  continuum  is  Euclidean.  Let 
A fx)  be  the  co-ordinates  of  the  vector  at  the  point  Pv 
Imagine  a  vector  drawn  at  the  point  Pv  using  the  local 
system  of  the  Xv,  with  the  same  co-ordinates  (parallel 
vector  through  P^)}  then  this  parallel  vector  is  uniquely 
determined  by  the  vector  at  P1  and  the  displacement. 
We  designate  this  operation,  whose  uniqueness  will  appear 
in  the  sequel,  the  parallel  displacement  of  the  vector  An 
from  P1  to  the  infinitesimally  near  point  P2  If  we  form 
the  vector  difference  of  the  vector  (A*)  at  the  point  P2 
and  the  vector  obtained  by  parallel  displacement  from  Px 
to  P2,  we  get  a  vector  which  may  be  regarded  as  the 
differential  of  the  vector  ( A for  the  given  displacement 
(dx,). 

This  vector  displacement  can  naturally  also  be  con¬ 
sidered  with  respect  to  the  co-ordinate  system  of  the  xv. 
If  Av  are  the  co-ordinates  of  the  vector  at  Plf  Av  +  &AV 
the  co-ordinates  of  the  vector  displaced  to  P2  along  the 
interval  (dxv),  then  the  SAU  do  not  vanish  in  this  case. 
We  know  of  these  quantities,  which  do  not  have  a  vector 


78  THE  MEANING  OF  RELATIVITY 


character,  that  they  must  depend  linearly  and  homo¬ 
geneously  upon  the  dxv  and  the  Av .  We  therefore  put 

SAV  =  -  T^A'dxp  .  .  (67) 

In  addition,  we  can  state  that  the  Tvap  must  be  sym¬ 
metrical  with  respect  to  the  indices  a  and  {3.  For  we 
can  assume  from  a  representation  by  the  aid  of  a  Euclid¬ 
ean  system  of  local  co-ordinates  that  the  same  parallelo¬ 
gram  will  be  described  by  the  displacement  of  an  element 
d[1)xv  along  a  second  element  d^xv  as  by  a  displacement 
of  d^xv  along  d^xv.  We  must  therefore  have 

d^\xv  +  (d[X)xv  -  T^K^Xp) 

=  d{1)xv  +  (d[~)xv  -  V^xj^xp). 

The  statement  made  above  follows  from  this,  after  inter¬ 
changing  the  indices  of  summation,  a  and  /3,  on  the 
right-hand  side. 

Since  the  quantities  g^v  determine  all  the  metrical 
properties  of  the  continuum,  they  must  also  determine 
the  T^.  If  we  consider  the  invariant  of  the  vector  Av, 
that  is,  the  square  of  its  magnitude, 

g^A” 

which  is  an  invariant,  this  cannot  change  in  a  parallel 
displacement.  We  therefore  have 

o  =  S(g^A»A')  =  jgA*A"dxa  +  g^A^SA*  +  g^ASA* 
or,  by  (67), 

-  g^ri  ~  g^K)^A”dxa  =  o. 


THE  GENERAL  THEORY  79 


Owing  to  the  symmetry  of  the  expression  in  the 
brackets  with  respect  to  the  indices  and  v ,  this  equation 
can  be  valid  for  an  arbitrary  choice  of  the  vectors  ( A a) 
and  dxv  only  when  the  expression  in  the  brackets  vanishes 
for  all  combinations  of  the  indices.  By  a  cyclic  inter¬ 
change  of  the  indices  fi,  v ,  a,  we  obtain  thus  altogether 
three  equations,  from  which  we  obtain,  on  taking  into 
account  the  symmetrical  property  of  the 

•  •  •  (68) 

in  which,  following  Christoffel,  the  abbreviation  has  been 
used, 


If  we  multiply  (68)  by  gacr  and  sum  over  the  a,  we 
obtain 


(70) 


in  which  {'7}  is  the  Christoffel  symbol  of  the  second 
kind.  Thus  the  quantities  T  are  deduced  from  the  g^v. 
Equations  (67)  and  (70)  are  the  foundation  for  the 
following  discussion. 

Co-variant  Differentiation  of  Tensors.  If  (A11  +  SAf  is 
the  vector  resulting  from  an  infinitesimal  parallel  displace¬ 
ment  from  P1  to  P 2,  and  ( A “  +  dA the  vector  A*  at  the 
point  P2l  then  the  difference  of  these  two, 


dA*  -  8A *  = 


+  Y^A^dX" 


80  THE  MEANING  OF  RELATIVITY 


is  also  a  vector.  Since  this  is  the  case  for  an  arbitrary 
choice  of  the  dxvi  it  follows  that 


A <7 


' dA fX 


(7 1) 


is  a  tensor,  which  we  designate  as  the  co-variant  derivative 
of  the  tensor  of  the  first  rank  (vector).  Contracting  this 
tensor,  we  obtain  the  divergence  of  the  contra-variant 
tensor  A\  In  this  we  must  observe  that  according  to 
(70), 


If  we  put,  further, 


()  gr 

_  JL  rr<ra  <7>  (Ta 

2a 


JL  h/A 

*Jg 


A*  Jg  =  B* 


(72) 

(73) 


a  quantity  designated  by  Weyl  as  the  contra-variant  tensor 
density  *  of  the  first  rank,  it  follows  that, 


^ 

is  a  scalar  density. 

We  get  the  law  of  parallel  displacement  for  the 
co-variant  vector  Z?  by  stipulating  that  the  parallel 
displacement  shall  be  effected  in  such  a  way  that  the 
scalar 


cf)  =  A^B^ 


remains  unchanged,  and  that  therefore 


Ar-ZBp  + 


*This  expression  is  justified,  in  that  Av-Jgdx  =  21  ^dx  has  a  tensor 
character.  Every  tensor,  when  multiplied  by  Jg,  changes  into  a  tensor 
density.  We  employ  capital  Gothic  letters  for  tensor  densities. 


THE  GENERAL  THEORY 


81 


vanishes  for  every  value  assigned  to  (A"-).  We  therefore 
get 

BBP  =  ri„AJx„.  .  .  .  (75) 

From  this  we  arrive  at  the  co-variant  derivative  of  the 
co-variant  vector  by  the  same  process  as  that  which  led 
to  (71), 

bp,  <T  -  ¥?*  -  r %bv  .  .  (76) 

ox9 

By  interchanging  the  indices  ^  and  a,  and  subtracting, 
we  get  the  skew-symmetrical  tensor, 


tea 


te>a 

te^ 


(77) 


For  the  co-variant  differentiation  of  tensors  of  the 
second  and  higher  ranks  we  may  use  the  process  by 
which  (75)  was  deduced.  Let,  for  example,  ( A ar)  be  a 
co- variant  tensor  of  the  second  rank.  Then  A^E^F7  is 
a  scalar,  if  E  and  F  are  vectors.  This  expression  must 
not  be  changed  by  the  8-displacement ;  expressing  this 
by  a  formula,  we  get,  using  (67),  SA aT,  whence  we  get  the 
desired  co-variant  derivative, 


A 


'bA 


CTT 


or;  p 


ten 


_  r,a  A  —  Pa  A 

A  ( Tp**-1  aT  A  Tp 


era • 


•  (78) 


In  order  that  the  general  law  of  co-variant  differ¬ 
entiation  of  tensors  may  be  clearly  seen,  we  shall  write 
down  two  co-variant  derivatives  deduced  in  an  analogous 
way : 


Al.a 

...  aA; 

a ,  p 

tep 

a*: 

'dA(TT 

.  p 

-  ra  AT  +  TT  Aa 

A  apxxa  *  A  apx±(T 

+  +  F'pA™. 


(79) 

(So) 


6 


82  THE  MEANING  OF  RELATIVITY 


The  general  law  of  formation  now  becomes  evident. 
From  these  formulae  we  shall  deduce  some  others  which 
are  of  interest  for  the  physical  applications  of  the  theory. 
In  case  Aar  is  skew-symmetrical,  we  obtain  the  tensor 


A 


arp 


+  3^  +  3  AfT 
~bxp  ~bx,  ixT 


(8 1 ) 


which  is  skew-symmetrical  in  all  pairs  of  indices,  by  cyclic 
interchange  and  addition. 

If,  in  (78),  we  replace  Aar  by  the  fundamental  tensor* 
gaT,  then  the  right-hand  side  vanishes  identically  ;  an 
analogous  statement  holds  for  (80)  with  respect  to  g aT ; 
that  is,  the  co-variant  derivatives  of  the  fundamental 
tensor  vanish.  That  this  must  be  so  we  see  directly  in 
the  local  system  of  co-ordinates. 

In  case  AaT  is  skew-symmetrical,  we  obtain  from  (80), 
by  contraction  with  respect  to  t  and  p, 


(82) 


In  the  general  case,  from  (79)  and  (80),  by  contraction 
with  respect  to  t  and  p,  we  obtain  the  equations, 


=  Tiffri.  .  .  (83) 

dXa 

a-  =  ^  .  .  (84) 

The  Riemann  Tensor .  If  we  have  given  a  curve  ex¬ 
tending  from  the  point  P  to  the  point  G  of  the  continuum, 
then  a  vector  A*,  given  at  P,  may,  by  a  parallel  displace¬ 
ment,  be  moved  along  the  curve  to  G.  If  the  continuum 


THE  GENERAL  THEORY 


83 


is  Euclidean  (more  generally,  if  by  a  suitable  choice  of 
co-ordinates  the^v  are  constants)  then  the  vector  obtained 
at  G  as  a  result  of  this  displacement  does  not  depend 
upon  the  choice  of  the  curve  joining  P  and  G.  But 
otherwise,  the  result  depends  upon  the  path  of  the  dis¬ 
placement.  In  this  case,  therefore,  a  vector  suffers  a 
change,  A A*  (in  its  direction,  not  its  magnitude),  when  it 
is  carried  from  a  point  P  of  a  closed  curve,  along  the 


Q 


curve,  and  back  to  P.  We  shall  now  calculate  this  vector 
change : 


A  A*  = 


As  in  Stokes’  theorem  for  the  line  integral  of  a  vector 
around  a  closed  curve,  this  problem  may  be  reduced  to 
the  integration  around  a  closed  curve  with  infinitely  small 
linear  dimensions ;  we  shall  limit  ourselves  to  this  case. 


84  THE  MEANING  OF  RELATIVITY 


We  have,  first,  by  (67), 


A  A*  = 


/» 

T%Aad*p 

a 

0 


In  this,  Tjjg  is  the  value  of  this  quantity  at  the  variable 
point  G  of  the  path  of  integration.  If  vve  put 


Z!L  ~  (xv)g  ~  (xv)p 

and  denote  the  value  of  Y^p  at  P  by  T^,  then  we  have, 
with  sufficient  accuracy, 

-  7nTva 

"PM  _  pM  1  UJ- 
L<*  “  ^  +  ■ 

Let,  further,  Aa  be  the  value  obtained  from  Aa  by  a 
parallel  displacement  along  the  curve  from  P  to  G.  It 
may  now  easily  be  proved  by  means  of  (67)  that  A M  -  A* 
is  infinitely  small  of  the  first  order,  while,  for  a  curve  of 
infinitely  small  dimensions  of  the  first  order,  A A*  is 
infinitely  small  of  the  second  order.  Therefore  there  is 
an  error  of  only  the  second  order  if  we  put 

Aa  =  ~A*  -  f lTA~aF- 

If  we  introduce  these  values  of  Y^p  and  Aa  into  the 
integral,  we  obtain,  neglecting  all  quantities  of  a  higher 
order  of  small  quantities  than  the  second, 


a  a*  =  -  gy  -  .  (85) 

0 

The  quantity  removed  from  under  the  sign  of  integration 


THE  GENERAL  THEORY 


85 


refers  to  the  point  P.  Subtracting  from  the 

integrand,  we  obtain 

o 

This  skew-symmetrical  tensor  of  the  second  rank,  faPi 
characterizes  the  surface  element  bounded  by  the  curve 
in  magnitude  and  position.  If  the  expression  in  the 
brackets  in  (85)  were  skew-symmetrical  with  respect  to 
the  indices  a  and  ft,  we  could  conclude  its  tensor  char¬ 
acter  from  (85).  We  can  accomplish  this  by  interchanging 
the  summation  indices  a  and  ft  in  (85)  and  adding  the 
resulting  equation  to  (85).  We  obtain 

2AA*  =  -  R\ *TmfiA*f+  .  .  (86) 

in  which 

ra  > 

+  W,  -  r&rk  (87) 


The  tensor  character  of  follows  from  (86) ;  this  is 
the  Riemann  curvature  tensor  of  the  fourth  rank,  whose 
properties  of  symmetry  we  do  not  need  to  go  into.  Its 
vanishing  is  a  sufficient  condition  (disregarding  the  reality 
of  the  chosen  co-ordinates)  that  the  continuum  is 
Euclidean. 

By  contraction  of  the  Riemann  tensor  with  respect  to 
the  indices  fi,  ft,  we  obtain  the  symmetrical  tensor  of  the 
second  rank, 


= 


ar;„ 


+  r^r? 


va 


+ 


ar;„ 


pa  p/3 
A  1 aj 3‘ 


(88) 


The  last  two  terms  vanish  if  the  system  of  co-ordinates 


86  THE  MEANING  OF  RELATIVITY 


is  so  chosen  that^  =  constant.  From  R^v  we  can  form 
the  scalar, 

R  =  •  •  •  (89) 


Straightest  ( Geodetic )  Lines.  A  line  may  be  constructed 
in  such  a  way  that  its  successive  elements  arise  from  each 
other  by  parallel  displacements.  This  is  the  natural 
generalization  of  the  straight  line  of  the  Euclidean 
geometry.  For  such  a  line,  we  have 


The  left-hand  side  is  to  be  replaced  by 


ds1 ’ 


so  that  we 


have 


+ 


dxa  dxp 
ds  ds 


o. 


(90) 


We  get  the  same  line  if  we  find  the  line  which  gives  a 
stationary  value  to  the  integral 


[ds  or  L/. 


g^dx^dx. 


between  two  points  (geodetic  line). 


*  The  direction  vector  at  a  neighbouring  point  of  the  curve  results,  by  a 
parallel  displacement  along  the  line  element  (<^^),  from  the  direction  vector 
of  each  point  considered. 


LECTURE  IV 


THE  GENERAL  THEORY  OF  RELATIVITY 

( Continued) 


WE  are  now  in  possession  of  the  mathematical 
apparatus  which  is  necessary  to  formulate  the 
laws  of  the  general  theory  of  relativity.  No  attempt 
will  be  made  in  this  presentation  at  systematic  complete¬ 
ness,  but  single  results  and  possibilities  will  be  devel¬ 
oped  progressively  from  what  is  known  and  from  the 
results  obtained.  Such  a  presentation  is  most  suited 
to  the  present  provisional  state  of  our  knowledge. 

A  material  particle  upon  which  no  force  acts  moves, 
according  to  the  principle  of  inertia,  uniformly  in  a 
straight  line.  In  the  four-dimensional  continuum  of  the 
special  theory  of  relativity  (with  real  time  co-ordinate) 
this  is  a  real  straight  line.  The  natural,  that  is,  the 
simplest,  generalization  of  the  straight  line  which  is 
plausible  in  the  system  of  concepts  of  Riemann’s  general 
theory  of  invariants  is  that  of  the  straightest,  or  geodetic, 
line.  We  shall  accordingly  have  to  assume,  in  the  sense 
of  the  principle  of  equivalence,  that  the  motion  of  a 
material  particle,  under  the  action  only  of  inertia  and 
gravitation,  is  described  by  the  equation, 


ds 2 


dxadx{ 3 
+  l^ds  ds 


87 


o. 


88  THE  MEANING  OF  RELATIVITY 


In  fact,  this  equation  reduces  to  that  of  a  straight  line 
if  all  the  components,  of  the  gravitational  field 

vanish. 

How  are  these  equations  connected  with  Newton’s 
equations  of  motion?  According  to  the  special  theory 
of  relativity,  the  g^v  as  well  as  the  g^v,  have  the  values, 
with  respect  to  an  inertial  system  (with  real  time  co¬ 
ordinate  and  suitable  choice  of  the  sign  of  ds 2), 

-  i  o  o 

o-i  o 

o  o  -  i 

ooo 


o 

o 

o 

i 


•  (91) 


The  equations  of  motion  then  become 


ds 2 


=  o. 


We  shall  call  this  the  “  first  approximation  ”  to  the  g!XV- 
field.  In  considering  approximations  it  is  often  useful, 
as  in  the  special  theory  of  relativity,  to  use  an  imaginary 
^-co-ordinate,  as  then  the  g fJLV,  to  the  first  approxima¬ 
tion,  assume  the  values 


(91a) 


These  values  may  be  collected  in  the  relation 

cr  —  —  $ 

&  fj-v  '-'txv' 

To  the  second  approximation  we  must  then  put 

S>y,v  —  “b  ’ 


(92) 


THE  GENERAL  THEORY 


89 


where  the  y^v  are  to  be  regarded  as  small  of  the  first 
order. 

Both  terms  of  our  equation  of  motion  are  then  small 
of  the  first  order.  If  we  neglect  terms  which,  relatively 
to  these,  are  small  of  the  first  order,  we  have  to  put 


1  /dyaj8 

2  \  'bxtL 


We  shall  now  introduce  an  approximation  of  a  second 
kind.  Let  the  velocity  of  the  material  particles  be  very 
small  compared  to  that  of  light.  Then  ds  will  be  the 

dxx  dx^  dx3 

same  as  the  time  differential,  dl.  Further, 

will  vanish  compared  to  We  shall  assume,  in  addi¬ 

tion,  that  the  gravitational  field  varies  so  little  with  the 
time  that  the  derivatives  of  the  y^v  by  xi  may  be 
neglected.  Then  the  equation  of  motion  (for  fi=  I,  2,  3) 
reduces  to 


d2x, 


V- 


dl1 


Lr^\  2  / 


(90a) 


This  equation  is  identical  with  Newton’s  equation  of 
motion  for  a  material  particle  in  a  gravitational  field,  if 

we  identify  with  the  potential  of  the  gravitational 

field  ;  whether  or  not  this  is  allowable,  naturally  depends 
upon  the  field  equations  of  gravitation,  that  is,  it  de¬ 
pends  upon  whether  or  not  this  quantity  satisfies,  to  a 
first  approximation,  the  same  laws  of  the  field  as  the 


90  THE  MEANING  OF  RELATIVITY 


gravitational  potential  in  Newton’s  theory.  A  glance 
at  (90)  and  (90a)  shows  that  the  Tjh  actually  do  play 
the  role  of  the  intensity  of  the  gravitational  field. 
These  quantities  do  not  have  a  tensor  character. 

Equations  (90)  express  the  influence  of  inertia  and 
gravitation  upon  the  material  particle.  The  unity  of 
inertia  and  gravitation  is  formally  expressed  by  the  fact 
that  the  whole  left-hand  side  of  (90)  has  the  character 
of  a  tensor  (with  respect  to  any  transformation  of  co¬ 
ordinates),  but  the  two  terms  taken  separately  do  not 
have  tensor  character,  so  that,  in  analogy  with  Newton’s 
equations,  the  first  term  would  be  regarded  as  the  ex¬ 
pression  for  inertia,  and  the  second  as  the  expression 
for  the  gravitational  force. 

We  must  next  attempt  to  find  the  laws  of  the  gravita¬ 
tional  field.  For  this  purpose,  Poisson’s  equation, 

A<£  =  \irKp 

of  the  Newtonian  theory  must  serve  as  a  model.  This 
equation  has  its  foundation  in  the  idea  that  the  gravi¬ 
tational  field  arises  from  the  density  p  of  ponderable 
matter.  It  must  also  be  so  in  the  general  theory  of 
relativity.  But  our  investigations  of  the  special  theory 
of  relativity  have  shown  that  in  place  of  the  scalar 
density  of  matter  we  have  the  tensor  of  energy  per  unit 
volume.  In  the  latter  is  included  not  only  the  tensor 
of  the  energy  of  ponderable  matter,  but  also  that  of  the 
electromagnetic  energy.  We  have  seen,  indeed,  that 
in  a  more  complete  analysis  the  energy  tensor  can  be 
regarded  only  as  a  provisional  means  of  representing 


THE  GENERAL  THEORY 


91 


matter.  In  reality,  matter  consists  of  electrically  charged 
particles,  and  is  to  be  regarded  itself  as  a  part,  in  fact, 
the  principal  part,  of  the  electromagnetic  field.  It  is 
only  the  circumstance  that  we  have  not  sufficient  know¬ 
ledge  of  the  electromagnetic  field  of  concentrated  charges 
that  compels  us,  provisionally,  to  leave  undetermined 
in  presenting  the  theory,  the  true  form  of  this  tensor. 
From  this  point  of  view  our  problem  now  is  to  introduce 
a  tensor,  T^,  of  the  second  rank,  whose  structure  we  do 
not  know  provisionally,  and  which  includes  in  itself  the 
energy  density  of  the  electromagnetic  field  and  of  ponder¬ 
able  matter ;  we  shall  denote  this  in  the  following  as 
the  “  energy  tensor  of  matter.” 

According  to  our  previous  results,  the  principles  of 
momentum  and  energy  are  expressed  by  the  statement 
that  the  divergence  of  this  tensor  vanishes  (47c).  In 
the  general  theory  of  relativity,  we  shall  have  to  assume 
as  valid  the  corresponding  general  co-variant  equation. 
If  (T^v)  denotes  the  co-variant  energy  tensor  of  matter, 
XKJ.  the  corresponding  mixed  tensor  density,  then,  in 
accordance  with  (83),  we  must  require  that 


o  = 


(95) 


be  satisfied.  It  must  be  remembered  that  besides  the 
energy  density  of  the  matter  there  must  also  be  given 
an  energy  density  of  the  gravitational  field,  so  that  there 
can  be  no  talk  of  principles  of  conservation  of  energy 
and  momentum  for  matter  alone.  This  is  expressed 
mathematically  by  the  presence  of  the  second  term  in 


92  THE  MEANING  OF  RELATIVITY 


(95),  which  makes  it  impossible  to  conclude  the  existence 
of  an  integral  equation  of  the  form  of  (49).  The  gravi¬ 
tational  field  transfers  energy  and  momentum  to  the 
“matter,”  in  that  it  exerts  forces  upon  it  and  gives  it 
energy;  this  is  expressed  by  the  second  term  in  (95). 

If  there  is  an  analogue  of  Poisson’s  equation  in  the 
general  theory  of  relativity,  then  this  equation  must  be 
a  tensor  equation  for  the  tensor  g^v  of  the  gravitational 
potential ;  the  energy  tensor  of  matter  must  appear  on 
the  right-hand  side  of  this  equation.  On  the  left-hand 
side  of  the  equation  there  must  be  a  differential  tensor 
in  the  g^v.  We  have  to  find  this  differential  tensor. 
It  is  completely  determined  by  the  following  three 
conditions : — 

1.  It  may  contain  no  differential  coefficients  of  the^ 
higher  than  the  second. 

2.  It  must  be  linear  and  homogeneous  in  these  second 
differential  coefficients. 

3.  Its  divergence  must  vanish  identically. 

The  first  two  of  these  conditions  are  naturally  taken 
from  Poisson’s  equation.  Since  it  may  be  proved 
mathematically  that  all  such  differential  tensors  can  be 
formed  algebraically  (i.e.  without  differentiation)  from 
Riemann’s  tensor,  our  tensor  must  be  of  the  form 

Kv  + 

in  which  R^v  and  R  are  defined  by  (88)  and  (89)  respec¬ 
tively.  Further,  it  may  be  proved  that  the  third  condi¬ 
tion  requires  a  to  have  the  value  -  For  the  law 


THE  GENERAL  THEORY  93 


of  the  gravitational  field  we  therefore  get  the  equa¬ 
tion 

Equation  (95)  is  a  consequence  of  this  equation,  tc  de¬ 
notes  a  constant,  which  is  connected  with  the  Newtonian 
gravitation  constant. 

In  the  following  I  shall  indicate  the  features  of  the 
theory  which  are  interesting  from  the  point  of  view  of 
physics,  using  as  little  as  possible  of  the  rather  involved 
mathematical  method.  It  must  first  be  shown  that  the 
divergence  of  the  left-hand  side  actually  vanishes.  The 
energy  principle  for  matter  may  be  expressed,  by  (83), 


0  -  £  - 


in  which 


Z",  =  - 


S- 


(97) 


The  analogous  operation,  applied  to  the  left-hand  side 
of  (96),  will  lead  to  an  identity. 

In  the  region  surrounding  each  world-point  there  are 
systems  of  co-ordinates  for  which,  choosing  the  ^^-co¬ 
ordinate  imaginary,  at  the  given  point, 


-  g*  =  _  =  o  if  ^  ={=  V) 

and  for  which  the  first  derivatives  of  the  g^v  and  the 
g*v  vanish.  We  shall  verify  the  vanishing  of  the  diverg¬ 
ence  of  the  left-hand  side  at  this  point.  At  this  point 
the  components  T^a  vanish,  so  that  we  have  to  prove 
the  vanishing  only  of 


94  THE  MEANING  OF  RELATIVITY 

Introducing  (88)  and  (70)  into  this  expression,  we  see 
that  the  only  terms  that  remain  are  those  in  which  third 
derivatives  of  the  g^v  enter.  Since  the  g  are  to  be 
replaced  by  -  we  obtain,  finally,  only  a  few  terms 
which  may  easily  be  seen  to  cancel  each  other.  Since 
the  quantity  that  we  have  formed  has  a  tensor  character, 
its  vanishing  is  proved  for  every  other  system  of  co-ordin¬ 
ates  also,  and  naturally  for  every  other  four-dimensional 
point.  The  energy  principle  of  matter  (97)  is  thus  a 
mathematical  consequence  of  the  field  equations  (96). 

In  order  to  learn  whether  the  equations  (96)  are 
consistent  with  experience,  we  must,  above  all  else,  find 
out  whether  they  lead  to  the  Newtonian  theory  as  a 
first  approximation.  For  this  purpose  we  must  intro¬ 
duce  various  approximations  into  these  equations.  We 
already  know  that  Euclidean  geometry  and  the  law  of  the 
constancy  of  the  velocity  of  light  are  valid,  to  a  certain 
approximation,  in  regions  of  a  great  extent,  as  in  the 
planetary  system.  If,  as  in  the  special  theory  of  rela¬ 
tivity,  we  take  the  fourth  co-ordinate  imaginary,  this 
means  that  we  must  put 

~  ~~  y^v  •  •  •  (9^) 

in  which  the  y^v  are  so  small  compared  to  1  that  we 
can  neglect  the  higher  powers  of  the  y ^  and  their 
derivatives.  If  we  do  this,  we  learn  nothing  about  the 
structure  of  the  gravitational  field,  or  of  metrical  space  of 
cosmical  dimensions,  but  we  do  learn  about  the  influence 
of  neighbouring  masses  upon  physical  phenomena. 

Before  carrying  through  this  approximation  we  shall 


THE  GENERAL  THEORY 


05 


transform  (96).  We  multiply  (96)  by  g*v,  summed  over 
the  fi  and  v ;  observing  the  relation  which  follows  from 
the  definition  of  the  g^, 

=  4 

we  obtain  the  equation 

R  =  /cgllvTflv  =  kT. 

If  we  put  this  value  of  R  in  (96)  we  obtain 

=  -  k{T^  -  =  -  tcTl,.  .  (96a) 

When  the  approximation  which  has  been  mentioned  is 
carried  out,  we  obtain  for  the  left-hand  side, 

+  'dxy^xv  dxyZxa  ^xg)xj 
or 

l3^-  ,  3  ■  3  va\ 

7  ix*  5.ra  /  2  J.ra  / 

in  which  has  been  put 

y  h-v  ~  y^v  ~  \y •  •  (99) 

We  must  now  note  that  equation  (96)  is  valid  for  any 
system  of  co-ordinates.  We  have  already  specialized  the 
system  of  co-ordinates  in  that  we  have  chosen  it  so  that 
within  the  region  considered  the  g^v  differ  infinitely  little 
from  the  constant  values  -  8^.  But  this  condition 
remains  satisfied  in  any  infinitesimal  change  of  co¬ 
ordinates,  so  that  there  are  still  four  conditions  to  which 
the  may  be  subjected,  provided  these  conditions  do 
not  conflict  with  the  conditions  for  the  order  of  magnitude 


96  THE  MEANING  OF  RELATIVITY 


of  the  y^.  We  shall  now  assume  that  the  system  of  co¬ 
ordinates  is  so  chosen  that  the  four  relations — 

_  l-i-v  ~^y  iav  i^ycrcr 

~  ~dXv  “  1xv  2  IXp. 
are  satisfied.  Then  (96a)  takes  the  form 

=  2 *t%  .  .  .  (96b) 

These  equations  may  be  solved  by  the  method,  familiar 
in  electrodynamics,  of  retarded  potentials;  we  get,  in  an 
easily  understood  notation, 


y  ixv 


f  9V  f  ~  r)JV 
27rJ  r  "  ( 


(101) 


In  order  to  see  in  what  sense  this  theory  contains  the 
Newtonian  theory,  we  must  consider  in  greater  detail 
the  energy  tensor  of  matter.  Considered  phenomeno¬ 
logically,  this  energy  tensor  is  composed  of  that  of  the 
electromagnetic  field  and  of  matter  in  the  narrower  sense. 
If  we  consider  the  different  parts  of  this  energy  tensor 
with  respect  to  their  order  of  magnitude,  it  follows 
from  the  results  of  the  special  theory  of  relativity  that 
the  contribution  of  the  electromagnetic  field  practically 
vanishes  in  comparison  to  that  of  ponderable  matter.  In 
our  system  of  units,  the  energy  of  one  gram  of  matter  is 
equal  to  I,  compared  to  which  the  energy  of  the  electric 
fields  may  be  ignored,  and  also  the  energy  of  deformation 
of  matter,  and  even  the  chemical  energy.  We  get  an 
approximation  that  is  fully  sufficient  for  our  purpose  if 


THE  GENERAL  THEORY  97 


we  put 

dx„  dxv  1 

^  ■*[ 

ds2  =  g^dxjx  J 


In  this,  <j  is  the  density  at  rest,  that  is,  the  density  of  the 
ponderable  matter,  in  the  ordinary  sense,  measured  with 
the  aid  of  a  unit  measuring  rod,  and  referred  to  a  Galilean 
system  of  co-ordinates  moving  with  the  matter. 

We  observe,  further,  that  in  the  co-ordinates  we  have 
chosen,  we  shall  make  only  a  relatively  small  error  if  we 
replace  the  g^v  by  -  8^,  so  that  we  put 


ds 2  =  -  ^dx2.  .  .  (102a) 

The  previous  developments  are  valid  however  rapidly 
the  masses  which  generate  the  field  may  move  relatively 
to  our  chosen  system  of  quasi-Galilean  co-ordinates.  But 
in  astronomy  we  have  to  do  with  masses  whose  velocities, 
relatively  to  the  co-ordinate  system  employed,  are  always 
small  compared  to  the  velocity  of  light,  that  is,  small 
compared  to  i,  with  our  choice  of  the  unit  of  time. 
We  therefore  get  an  approximation  which  is  sufficient 
for  nearly  all  practical  purposes  if  in  (ioi)  we  replace 
the  retarded  potential  by  the  ordinary  (non-retarded) 
potential,  and  if,  for  the  masses  which  generate  the  field, 
we  put 


_  dx.j,  dx 3  dx±  ./  —  \dl  - 

ds  ds  ~  ds  ~  | ds  ~  =  ^  ~  I’  (I03a) 


7 


98  THE  MEANING  OF  RELATIVITY 


Then  we  get  for  and  Tnv  the  values 


o 

o 

o 

o 


o  o 

o  o 

o  o 

o  o 


o 

o 

o 

<J 


l 

I 


(104) 


For  T  we  get  the  value  cr,  and,  finally,  for  T*„  the 
values, 


a 

2 

o 

o 


o 

<7 

2 

O 


o  o 

We  thus  get,  from  (10 1), 
7ll  =  722  =  733  = 


o 

o 

cr 

2 

O  - 


O 

O 

O 

o  I 
2J 


744  =  + 


K 

'odVQ\ 

47 T. 

r 

K 

m<rdV0 

477. 

r  J 

(104a) 


(101a) 


while  all  the  other  y^v  vanish.  The  least  of  these  equa¬ 
tions,  in  connexion  with  equation  (90a),  contains  New¬ 
ton’s  theory  of  gravitation.  If  we  replace  /  by  ct  we 

get 

drx  kc 2  7)  ff odV0[ 

~df  ~  J 


We  see  that  the  Newtonian  gravitation  constant  W,  is 
connected  with  the  constant  tc  that  enters  into  our  field 
equations  by  the  relation 


K  = 


fCC1 

877' 


(105) 


99 


THE  GENERAL  THEORY 


From  the  known  numerical  value  of  K ,  it  therefore 
follows  that 


k  = 


877 K  877 . 6-67  .  10 


-8 


r 


9  .  10 


20 


1  -86  .  1  o~27.  (105a) 


From  (10 1 )  we  see  that  even  in  the  first  approximation 
the  structure  of  the  gravitational  field  differs  fundamentally 
from  that  which  is  consistent  with  the  Newtonian  theory  ; 
this  difference  lies  in  the  fact  that  the  gravitational 
potential  has  the  character  of  a  tensor  and  not  a  scalar. 
This  was  not  recognized  in  the  past  because  only  the 
component  g 44,  to  a  first  approximation,  enters  the  equa¬ 
tions  of  motion  of  material  particles. 

In  order  now  to  be  able  to  judge  the  behaviour  of 
measuring  rods  and  clocks  from  our  results,  we  must 
observe  the  following.  According  to  the  principle  of 
equivalence,  the  metrical  relations  of  the  Euclidean 
geometry  are  valid  relatively  to  a  Cartesian  system  of 
reference  of  infinitely  small  dimensions,  and  in  a  suitable 
state  of  motion  (freely  failing,  and  without  rotation). 
We  can  make  the  same  statement  for  local  systems  of 
co-ordinates  which,  relatively  to  these,  have  small  ac¬ 
celerations,  and  therefore  for  such  systems  of  co-ordinates 
as  are  at  rest  relatively  to  the  one  we  have  selected.  For 
such  a  local  system,  we  have,  for  two  neighbouring  point 
events, 

ds2  =  -  dX2  -  dX2  -  dX2  +  dT2  =  -  dS 2  +  dT2 

where  dS  is  measured  directly  by  a  measuring  rod  and 
dT  by  a  clock  at  rest  relatively  to  the  system  :  these  are 


100  THE  MEANING  OF  RELATIVITY 


the  naturally  measured  lengths  and  times.  Since  ds\  on 
the  other  hand,  is  known  in  terms  of  the  co-ordinates  xv 
employed  in  finite  regions,  in  the  form 


ds-  =  g^dx^dx. 


we  have  the  possibility  of  getting  the  relation  between 
naturally  measured  lengths  and  times,  on  the  one  hand, 
and  the  corresponding  differences  of  co-ordinates,  on  the 
other  hand.  As  the  division  into  space  and  time  is  in 
agreement  with  respect  to  the  two  systems  of  co-ordinates, 
so  when  we  equate  the  two  expressions  for  ds2  we  get 
two  relations.  If,  by  (ioia),  we  put 


we  obtain,  to  a  sufficiently  close  approximation, 


(i°6) 


The  unit  measuring  rod  has  therefore  the  length, 


in  respect  to  the  system  of  co-ordinates  we  have  selected. 
The  particular  system  of  co-ordinates  we  have  selected 


THE  GENERAL  THEORY 


101 


insures  that  this  length  shall  depend  only  upon 
the  place,  and  not  upon  the  direction.  If  we  had 
chosen  a  different  system  of  co-ordinates  this  would  not 
be  so.  But  however  we  may  choose  a  system  of  co¬ 
ordinates,  the  laws  of  configuration  of  rigid  rods  do  not 
agree  with  those  of  Euclidean  geometry  ;  in  other  words, 
we  cannot  choose  any  system  of  co-ordinates  so  that  the 
co-ordinate  differences,  Axl}  Ax2,  Axs,  corresponding  to  the 
ends  of  a  unit  measuring  rod,  oriented  in  any  way,  shall 
always  satisfy  the  relation  Ax}  +  Ax}  +  Ax-}  =  i.  In 
this  sense  space  is  not  Euclidean,  but  “  curved. ”  It 
follows  from  the  second  of  the  relations  above  that  the 
interval  between  two  beats  of  the  unit  clock  ( dT  =  i) 
corresponds  to  the  “  time  ” 


in  the  unit  used  in  our  system  of  co-ordinates.  The  rate 
of  a  clock  is  accordingly  slower  the  greater  is  the  mass  of 
the  ponderable  matter  in  its  neighbourhood.  We  there¬ 
fore  conclude  that  spectral  lines  which  are  produced  on 
the  sun’s  surface  will  be  displaced  towards  the  red, 
compared  to  the  corresponding  lines  produced  on  the 
earth,  by  about  2.  io~°  of  their  wave-lengths.  At  first, 
this  important  consequence  of  the  theory  appeared  to 
conflict  with  experiment ;  but  results  obtained  during  the 
past  year  seem  to  make  the  existence  of  this  effect  more 
probable,  and  it  can  hardly  be  doubted  that  this  con¬ 
sequence  of  the  theory  will  be  confirmed  within  the  next 
year. 


102  THE  MEANING  OF  RELATIVITY 


Another  important  consequence  of  the  theory,  which 
can  be  tested  experimentally,  has  to  do  with  the  path  of 
rays  of  light.  In  the  general  theory  of  relativity  also 
the  velocity  of  light  is  everywhere  the  same,  relatively  to 
a  local  inertial  system.  This  velocity  is  unity  in  our 
natural  measure  of  time.  The  law  of  the  propagation  of 
light  in  general  co-ordinates  is  therefore,  according  to  the 
general  theory  of  relativity,  characterized,  by  the  equation 

ds2  =  o. 


To  within  the  approximation  which  we  are  using,  and  in 
the  system  of  co-ordinates  which  we  have  selected,  the 
velocity  of  light  is  characterized,  according  to  (106),  by 
the  equation 


( 


I  + 


q*  dx<£  q-  dxd 


The  velocity  of  light  A,  is  therefore  expressed  in  our 
co-ordinates  by 


v/  dx2  q-  dx2  q-  dx2 
~dl 


k  [crdVr,  ,  v 

—  — -*•  (I07) 

47tJ  r 


We  can  therefore  draw  the  conclusion  from  this,  that  a 
ray  of  light  passing  near  a  large  mass  is  deflected.  If 
we  imagine  the  sun,  of  mass  M ,  concentrated  at  the 
origin  of  our  system  of  co-ordinates,  then  a  ray  of  light, 
travelling  parallel  to  the  ^3-axis,  in  the  x1  -  xs  plane, 
at  a  distance  A  from  the  origin,  will  be  deflected,  in  all, 
by  an  amount 


THE  GENERAL  THEORY 


103 


+  » 

f 1  , 

a  =  —  —ax^ 

JL  dxx 


towards  the  sun. 


On  performing  the  integration  we  get 

kM 


a 


27tA 


(ioS) 


The  existence  of  this  deflection,  which  amounts  to 
i  7 "  for  A  equal  to  the  radius  of  the  sun,  was  confirmed, 
with  remarkable  accuracy,  by  the  English  Solar  Eclipse 
.  Expedition  in  1919,  and  most  careful  preparations  have 
been  made  to  get  more  exact  observational  data  at  the 
solar  eclipse  in  1922.  It  should  be  noted  that  this 
result,  also,  of  the  theory  is  not  influenced  by  our 
arbitrary  choice  of  a  system  of  co-ordinates. 

This  is  the  place  to  speak  of  the  third  consequence  of 
the  theory  which  can  be  tested  by  observation,  namely, 
that  which  concerns  the  motion  of  the  perihelion 
of  the  planet  Mercury.  The  secular  changes  in  the 
planetary  orbits  are  known  with  such  accuracy  that  the 
approximation  we  have  been  using  is  no  longer  sufficient 
for  a  comparison  of  theory  and  observation.  It  is  neces¬ 
sary  to  go  back  to  the  general  field  equations  (96).  To 
solve  this  problem  I  made  use  of  the  method  of  succes¬ 
sive  approximations.  Since  then,  however,  the  problem 
of  the  central  symmetrical  statical  gravitational  field  has 
been  completely  solved  by  Schwarzschild  and  others ; 
the  derivation  given  by  H.  Weyl  in  his  book,  “  Raum- 
Zeit-Materie,”  is  particularly  elegant.  The  calculation 
can  be  simplified  somewhat  if  we  do  not  go  back  directly 


104  THE  MEANING  OF  RELATIVITY 


to  the  equation  (96),  but  base  it  upon  a  principle  of 
variation  that  is  equivalent  to  this  equation.  I  shall 
indicate  the  procedure  only  in  so  far  as  is  necessary  for 
understanding  the  method. 

In  the  case  of  a  statical  field,  ds2  must  have  the  form 

1 ds2  =  -  dc r2  +  f2dx± 

da-  =  ^Yapdxjxp 
1-3 

where  the  summation  on  the  right-hand  side  of  the  last 
equation  is  to  be  extended  over  the  space  variables  only, 
The  central  symmetry  of  the  field  requires  the  y^v  to  be 
of  the  form, 

Ya£  =  /^a/3  +  '^X0X) 3  *  *  ( 1  1  °) 

f  2,  n  and  \  are  functions  of  r  —  ^/x2  +  x£  +  xz2  only. 
One  of  these  three  functions  can  be  chosen  arbitrarily, 
because  our  system  of  co-ordinates  is,  a  priori ,  completely 
arbitrary  ;  for  by  a  substitution 

^4  =  *4 

V«  =  F(r)xa 

we  can  always  insure  that  one  of  these  three  functions 
shall  be  an  assigned  function  of  r.  In  place  of  (i  io)  we 
can  therefore  put,  without  limiting  the  generality, 

7a£  =  ^  a  8  +  .  .  (uoa) 

In  this  way  the  g^v  are  expressed  in  terms  of  the  two 
quantities  \  and  f.  These  are  to  be  determined  as  func¬ 
tions  of  r,  by  introducing  them  into  equation  (96),  after 


THE  GENERAL  THEORY 


105 


first  calculating  the  from  (107)  and  (108a).  We 
have 


-L  afi 


r4 

1 44 

r4 

1  4a 


+  2\r8ap 


—  u 


r  I+^r-^(fora,A^  =  1,2,3) 
r“3  =  =  o  (for  a,  /3  =  I,  2,  3) 

-2y2 


—  if 


=  -  */ 


-2¥! 


(108b) 


With  the  help  of  these  results,  the  field  equations 
furnish  Schwarzschild’s  solution  : 


ds2  = 


“  dr2 
A 

1 - 

r 


4-  r2(sin2  0dcf)2  +  d02) 


in  which  we  have  put 


(109) 


xA  =  l 

4 

xY  =  r  sin  6  sin  (p 
x.2  =  r  sin  0  cos  </> 
xz  =  r  cos  0 


A  = 


kM 

47 r 


(109a) 


M  denotes  the  sun’s  mass,  centrally  symmetrically 
placed  about  the  origin  of  co-ordinates  ;  the  solution  (109) 
is  valid  only  outside  of  this  mass,  where  all  the  T^v  vanish. 
If  the  motion  of  the  planet  takes  place  in  the  x1  -  x.2 
plane  then  we  must  replace  (109)  by 

/  A  \  dv2 

ds 2  —  [1  -  -yjdl2  -  - ^  -  r2d(p2  .  (109b) 

1  -  — 
r 


106  THE  MEANING  OF  RELATIVITY 


The  calculation  of  the  planetary  motion  depends  upon 
equation  (90).  From  the  first  of  equations  (108b)  and 
(90)  we  get,  for  the  indices  1,  2,  3, 

d  (  dxp  dx  \ 
ds\X * ds  ~  x*ds)  ~° 


or,  if  we  integrate,  and  express  the  result  in  polar  co¬ 
ordinates, 


d(p 

r =  constant. 


(no 


From  (90),  for  jj,  =  4,  we  get 

dr  l  I  df 2  dxa  dr  l  I  df~ 
0  ds 2  +  f'1  dxa  ds  ~  ds 2  +  f2  ds' 


From  this,  after  multiplication  by /2  and  integration,  we 

have 


=  constant. 


(1 1 2) 


In  (109b),  (ill)  and  (112)  we  have  three  equations 
between  the  four  variables  j,  r,  /  and  </>,  from  which  the 
motion  of  the  planet  may  be  calculated  in  the  same  way 
as  in  classical  mechanics.  The  most  important  result  we 
get  from  this  is  a  secular  rotation  of  the  elliptic  orbit  of 
the  planet  in  the  same  sense  as  the  revolution  of  the 
planet,  amounting  in  radians  per  revolution  to 


THE  GENERAL  THEORY  107 


where 

a  =  the  semi-major  axis  of  the  planetary  orbit  in 
centimetres. 

e  =  the  numerical  eccentricity. 

c  =  3  .  io+10,  the  velocity  of  light  in  vacuo . 

T  =  the  period  of  revolution  in  seconds. 

This  expression  furnishes  the  explanation  of  the  motion 
of  the  perihelion  of  the  planet  Mercury,  which  has  been 
known  for  a  hundred  years  (since  Leverrier),  and  for 
which  theoretical  astronomy  has  hitherto  been  unable 
satisfactorily  to  account. 

There  is  no  difficulty  in  expressing  Maxwell’s  theory 
of  the  electromagnetic  field  in  terms  of  the  general  theory 
of  relativity ;  this  is  done  by  application  of  the  tensor 
formation  (8 1 ),  (82)  and  (77).  Let  (p ^  be  a  tensor  of  the 
first  rank,  to  be  denoted  as  an  electromagnetic  4-potential ; 
then  an  electromagnetic  field  tensor  may  be  defined  by 
the  relations, 


^<j>u 

He’ 


(”4) 


The  second  of  Maxwell’s  systems  of  equations  is  then 
defined  by  the  tensor  equation,  resulting  from  this, 


T>x, 


(114a) 


and  the  first  of  Maxwell’s  systems  of  equations  is  defined 
by  the  tensor-density  relation 


108  THE  MEANING  OF  RELATIVITY 


in  which 


fl*”  =  V  -  gg^g1^ 

J  s  SPds ■ 


(XT 


If  we  introduce  the  energy  tensor  of  the  electromagnetic 
field  into  the  right-hand  side  of  (96),  we  obtain  (1 1 5), 
for  the  special  case  3^  =  o,  as  a  consequence  of  (96)  by 
taking  the  divergence.  This  inclusion  of  the  theory  of 
electricity  in  the  scheme  of  the  general  theory  of  relativity 
has  been  considered  arbitrary  and  unsatisfactory  by 
many  theoreticians.  Nor  can  we  in  this  way  conceive  of 
the  equilibrium  of  the  electricity  which  constitutes  the 
elementary  electrically  charged  particles.  A  theory  in 
which  the  gravitational  field  and  the  electromagnetic  field 
enter  as  an  essential  entity  would  be  much  preferable. 
H.  Weyl,  and  recently  Th.  Kaluza,  have  discovered  some 
ingenious  theorems  along  this  direction ;  but  concerning 
them,  I  am  convinced  that  they  do  not  bring  us  nearer  to 
the  true  solution  of  the  fundamental  problem.  I  shall 
not  go  into  this  further,  but  shall  give  a  brief  discussion 
of  the  so-called  cosmological  problem,  for  without  this, 
the  considerations  regarding  the  general  theory  of  rela¬ 
tivity  would,  in  a  certain  sense,  remain  unsatisfactory. 

Our  previous  considerations,  based  upon  the  field 
equations  (96),  had  for  a  foundation  the  conception  that 
space  on  the  whole  is  Galilean-Euclidean,  and  that  this 
character  is  disturbed  only  by  masses  embedded  in  it. 
This  conception  was  certainly  justified  as  long  as  we  were 
dealing  with  spaces  of  the  order  of  magnitude  of  those 


THE  GENERAL  THEORY 


109 


that  astronomy  has  to  do  with.  But  whether  portions  of 
the  universe,  however  large  they  may  be,  are  quasi- 
Euclidean,  is  a  wholly  different  question.  We  can  make 
this  clear  by  using  an  example  from  the  theory  of  surfaces 
which  we  have  employed  many  times.  If  a  portion  of  a 
surface  is  observed  by  the  eye  to  be  practically  plane,  it 
does  not  at  all  follow  that  the  whole  surface  has  the  form 
of  a  plane  ;  the  surface  might  just  as  well  be  a  sphere,  for 
example,  of  sufficiently  large  radius.  The  question  as  to 
whether  the  universe  as  a  whole  is  non-Euclidean  was 
much  discussed  from  the  geometrical  point  of  view  before 
the  development  of  the  theory  of  relativity.  But  with  the 
theory  of  relativity,  this  problem  has  entered  upon  a 
new  stage,  for  according  to  this  theory  the  geometrical 
properties  of  bodies  are  not  independent,  but  depend 
upon  the  distribution  of  masses. 

If  the  universe  were  quasi-Euclidean,  then  Mach  was 
wholly  wrong  in  his  thought  that  inertia,  as  well  as 
gravitation,  depends  upon  a  kind  of  mutual  action  between 
bodies.  For  in  this  case,  with  a  suitably  selected  system 
of  co-ordinates,  the  g^v  would  be  constant  at  infinity,  as 
they  are  in  the  special  theory  of  relativity,  while  within 
finite  regions  the  gixv  would  differ  from  these  constant 
values  by  small  amounts  only,  with  a  suitable  choice  of 
co-ordinates,  as  a  result  of  the  influence  of  the  masses  in 
finite  regions.  The  physical  properties  of  space  would 
not  then  be  wholly  independent,  that  is,  uninfluenced  by 
matter,  but  in  the  main  they  would  be,  and  only  in 
small  measure,  conditioned  by  matter.  Such  a  dualistic 
conception  is  even  in  itself  not  satisfactory;  there  are, 


110  THE  MEANING  OF  RELATIVITY 


however,  some  important  physical  arguments  against  it, 
which  we  shall  consider. 

The  hypothesis  that  the  universe  is  infinite  and 
Euclidean  at  infinity,  is,  from  the  relativistic  point  of 
view,  a  complicated  hypothesis.  In  the  language  of  the 
general  theory  of  relativity  it  demands  that  the  Riemann 
tensor  of  the  fourth  rank  R^imi  shall  vanish  at  infinity, 
which  furnishes  twenty  independent  conditions,  while  only 
ten  curvature  components  R  }  enter  into  the  laws  of  the 
gravitational  field.  It  is  certainly  unsatisfactory  to 
postulate  such  a  far-reaching  limitation  without  any 
physical  basis  for  it. 

But  in  the  second  place,  the  theory  of  relativity  makes 
it  appear  probable  that  Mach  was  on  the  right  road  in 
his  thought  that  inertia  depends  upon  a  mutual  action  of 
matter.  For  we  shall  show  in  the  following  that,  accord¬ 
ing  to  our  equations,  inert  masses  do  act  upon  each  other 
in  the  sense  of  the  relativity  of  inertia,  even  if  only  very 
feebly.  What  is  to  be  expected  along  the  line  of  Mach’s 
thought  ? 

1.  The  inertia  of  a  body  must  increase  when  ponder¬ 

able  masses  are  piled  up  in  its  neighbourhood. 

2.  A  body  must  experience  an  accelerating  force  when 

neighbouring  masses  are  accelerated,  and,  in  fact, 
the  force  must  be  in  the  same  direction  as  the 
acceleration. 

3.  A  rotating  hollow  body  must  generate  inside  of 

itself  a  “  Coriolis  field,”  which  deflects  moving 
bodies  in  the  sense  of  the  rotation,  and  a  radial 
centrifugal  field  as  well. 


THE  GENERAL  THEORY 


111 


We  shall  now  show  that  these  three  effects,  which  are 
to  be  expected  in  accordance  with  Mach’s  ideas,  are 
actually  present  according  to  our  theory,  although  their 
magnitude  is  so  small  that  confirmation  of  them  by 
laboratory  experiments  is  not  to  be  thought  of.  For  this 
purpose  we  shall  go  back  to  the  equations  of  motion  of 
a  material  particle  (90),  and  carry  the  approximations 
somewhat  further  than  was  done  in  equation  (90a). 

First,  we  consider  y4l  as  small  of  the  first  order.  The 
square  of  the  velocity  of  masses  moving  under  the  influence 
of  the  gravitational  force  is  of  the  same  order,  according 
to  the  energy  equation.  It  is  therefore  logical  to  regard 
the  velocities  of  the  material  particles  we  are  considering, 
as  well  as  the  velocities  of  the  masses  which  generate  the 
field,  as  small,  of  the  order  -J.  We  shall  now  carry  out  the 
approximation  in  the  equations  that  arise  from  the  field 
equations  (101)  and  the  equations  of  motion  (90)  so  far 
as  to  consider  terms,  in  the  second  member  of  (90),  that 
are  linear  in  those  velocities.  Further,  we  shall  not  put 
ds  and  dl  equal  to  each  other,  but,  corresponding  to  the 
higher  approximation,  we  shall  put 

ds  =  JFJi  =  0  ~ 


From  (90)  we  obtain,  at  first, 


+ 


744\^ 

2  )  dl  J 


TV* 
1  a/3 


dx„  dx 


dl 


a  dxj  y44\  , 

+  fMIl6> 


From  (1 01)  we  get,  to  the  approximation  sought  for, 


112  THE  MEANING  OF  RELATIVITY 


Yu 


722  = 


733  =  744 

K  |  o 

47rJ 

i  x 

C  dx0 

74a  =  - 

2 

G  ds 

Y«0  =  0 

J 

/ 

r 


(i  17) 


in  which,  in  (1 17),  a  and  /3  denote  the  space  indices  only. 
On  the  right-hand  side  of  (116)  we  can  replace 

r/ 

1  +2 'by  1  and  -  I7  by  [“/].  It  is  easy  to  see,  in 

addition,  that  to  this  degree  of  approximation  we  must 
put 

M  = 


ail  _  1  P>V* 


[;4]  - 
[f]  -  ° 


4a 


cXtr 


'bx, 


p- 1 


in  which  a,  /3  and  fi  denote  space  indices.  We  therefore 
obtain  from  (116),  in  the  usual  vector  notation, 


d  _  ^B 

.1  +  <J>]  =  grad  a-  +  +  [rot  B,  v] 


k  |WF0 


H  = 


8ttJ  r 

K  r«%dv. 


\.  (1 18) 


dl 


0 


The  equations  of  motion,  (i  1 8),  show  now,  in  fact,  that 


THE  GENERAL  THEORY  113 


1.  The  inert  mass  is  proportional  to  I  +  a,  and 

therefore  increases  when  ponderable  masses 
approach  the  test  body. 

2.  There  is  an  inductive  action  of  accelerated  masses, 

of  the  same  sign,  upon  the  test  body.  This  is 

,  m 

the  term 

3.  A  material  particle,  moving  perpendicularly  to  the 

axis  of  rotation  inside  a  rotating  hollow  body, 
is  deflected  in  the  sense  of  the  rotation  (Coriolis 
field).  The  centrifugal  action,  mentioned  above, 
inside  a  rotating  hollow  body,  also  follows  from 
the  theory,  as  has  been  shown  by  Thirring.* 

Although  all  of  these  effects  are  inaccessible  to  experi¬ 
ment,  because  k  is  so  small,  nevertheless  they  certainly 
exist  according  to  the  general  theory  of  relativity.  We 
must  see  in  them  a  strong  support  for  Mach’s  ideas  as  to 
the  relativity  of  all  inertial  actions.  If  we  think  these 
ideas  consistently  through  to  the  end  we  must  expect  the 
whole  inertia,  that  is,  the  whole  ^-field,  to  be  determined 
by  the  matter  of  the  universe,  and  not  mainly  by  the 
boundary  conditions  at  infinity. 

For  a  satisfactory  conception  of  the  ^,,-field  of  cosmical 
dimensions,  the  fact  seems  to  be  of  significance  that  the 
relative  velocity  of  the  stars  is  small  compared  to  the 
velocity  of  light.  It  follows  from  this  that,  with  a  suit- 

*  That  the  centrifugal  action  must  be  inseparably  connected  with  the 
existence  of  the  Coriolis  field  may  be  recognized,  even  without  calculation, 
in  the  special  case  of  a  co-ordinate  system  rotating  uniformly  relatively  to 
an  inertial  system  ;  our  general  co-variant  equations  naturally  must  apply 
to  such  a  case. 

8 


114  THE  MEANING  OF  RELATIVITY 


able  choice  of  co-ordinates,  gu  is  nearly  constant  in  the 
universe,  at  least,  in  that  part  of  the  universe  in  which 
there  is  matter.  The  assumption  appears  natural,  more¬ 
over,  that  there  are  stars  in  all  parts  of  the  universe,  so 
that  we  may  well  assume  that  the  inconstancy  of  g^ 
depends  only  upon  the  circumstance  that  matter  is  not 
distributed  continuously,  but  is  concentrated  in  single 
celestial  bodies  and  systems  of  bodies.  If  we  are  willing 
to  ignore  these  more  local  non-uniformities  of  the  density 
of  matter  and  of  the  ^-field,  in  order  to  learn  something 
of  the  geometrical  properties  of  the  universe  as  a  whole, 
it  appears  natural  to  substitute  for  the  actual  distribution 
of  masses  a  continuous  distribution,  and  furthermore  to 
assign  to  this  distribution  a  uniform  density  a.  In  this 
imagined  universe  all  points  with  space  directions  will 
be  geometrically  equivalent ;  with  respect  to  its  space 
extension  it  will  have  a  constant  curvature,  and  will  be 
cylindrical  with  respect  to  its  ^4-co-ordinate.  The  pos¬ 
sibility  seems  to  be  particularly  satisfying  that  the  universe 
is  spatially  bounded  and  thus,  in  accordance  with  our 
assumption  of  the  constancy  of  a,  is  of  constant  curvature, 
being  either  spherical  or  elliptical ;  for  then  the  boundary 
conditions  at  infinity  which  are  so  inconvenient  from  the 
standpoint  of  the  general  theory  of  relativity,  may  be 
replaced  by  the  much  more  natural  conditions  for  a  closed 
surface. 

According  to  what  has  been  said,  we  are  to  put 

ds1  =  dx£  -  7 [).vdxgixv  .  .  (i  1 9) 

in  which  the  indices  fi  and  v  run  from  1  to  3  only.  The 


115 


THE  GENERAL  THEORY 


7M„  will  be  such  functions  of  xx,  x 2,  xz  as  correspond 
to  a  three-dimensional  continuum  of  constant  positive 
curvature.  We  must  now  investigate  whether  such  an 
assumption  can  satisfy  the  field  equations  of  gravitation. 

In  order  to  be  able  to  investigate  this,  we  must  first 
find  what  differential  conditions  the  three-dimensional 
manifold  of  constant  curvature  satisfies.  A  spherical 
manifold  of  three  dimensions,  embedded  in  a  Euclidean 
continuum  of  four  dimensions,*  is  given  by  the  equations 


x\  +  x2  +  x./  +  x2  =  a 2 
dx^  +  dx o2  +  dxd  +  dx2  =  ds1. 


By  eliminating  xi}  we  get 
ds1  —  dx±  +  dx2  +  dx22  + 


(x1dxl  +  x2dx2  +  x-^dxj2 
d 2  -  X2  -  x2  -  x 2 


As  far  as  terms  of  the  third  and  higher  degrees  in  the 
xv,  we  can  put,  in  the  neighbourhood  of  the  origin  of 
co-ordinates, 

ds1  =  (s„,  +  X-^)dxvdxv. 


Inside  the  brackets  are  the  g^v  of  the  manifold  in  the 
neighbourhood  of  the  origin.  Since  the  first  derivatives 
of  the  g^v,  and  therefore  also  the  Y vanish  at  the 
origin,  the  calculation  of  the  R^v  for  this  manifold,  by 
(88),  is  very  simple  at  the  origin.  We  have 


2  <N  _  2 

5  -  jgpr 


*  The  aid  of  a  fourth  space  dimension  has  naturally  no  significance 
except  that  of  a  mathematical  artifice. 


116  THE  MEANING  OF  RELATIVITY 


2 

Since  the  relation  is  universally  co-variant, 

and  since  all  points  of  the  manifold  are  geometrically 
equivalent,  this  relation  holds  for  every  system  of  co¬ 
ordinates,  and  everywhere  in  the  manifold.  In  order  to 
avoid  confusion  with  the  four-dimensional  continuum, 
we  shall,  in  the  following,  designate  quantities  that  refer 
to  the  three-dimensional  continuum  by  Greek  letters, 
and  put 

2 

P,uv  =  —  -tfnv  •  •  •  (i  20) 

We  now  proceed  to  apply  the  field  equations  (96)  to 
our  special  case.  From  (1 19)  we  get  for  the  four-dimen¬ 
sional  manifold, 


R^v  =  PM„  for  the  indices  1  to  3 

^14  =  ^24  =  ^34  =  ^44  =  O 


(121) 


For  the  right-hand  side  of  (96)  we  have  to  consider 
the  energy  tensor  for  matter  distributed  like  a  cloud  of 
dust.  According  to  what  has  gone  before  we  must 
therefore  put 


T>xv  = 


dx,L  dxv 
a — -  • — 
ds  ds 


specialized  for  the  case  of  rest.  But  in  addition,  we 
shall  add  a  pressure  term  that  may  be  physically  estab¬ 
lished  as  follows.  Matter  consists  of  electrically  charged 
particles.  On  the  basis  of  Maxwell’s  theory  these 
cannot  be  conceived  of  as  electromagnetic  fields  free 
from  singularities.  In  order  to  be  consistent  with  the 


THE  GENERAL  THEORY 


117 


facts,  it  is  necessary  to  introduce  energy  terms,  not  con¬ 
tained  in  Maxwell’s  theory,  so  that  the  single  electric 
particles  may  hold  together  in  spite  of  the  mutual  re¬ 
pulsions  between  their  elements,  charged  with  electricity 
of  one  sign.  For  the  sake  of  consistency  with  this  fact, 
Poincare  has  assumed  a  pressure  to  exist  inside  these 
particles  which  balances  the  electrostatic  repulsion.  It 
cannot,  however,  be  asserted  that  this  pressure  vanishes 
outside  the  particles.  We  shall  be  consistent  with  this 
circumstance  if,  in  our  phenomenological  presentation, 
we  add  a  pressure  term.  This  must  not,  however,  be 
confused  with  a  hydrodynamical  pressure,  as  it  serves 
only  for  the  energetic  presentation  of  the  dynamical 
relations  inside  matter.  In  this  sense  we  put 


^  axa  ax  b  ,  \ 

T H.V  ~  ~  g^p.  • 


In  our  special  case  we  have,  therefore,  to  put 


T^v  =  y^p  (for  and  v  from  1  to  3) 


!  =  -  y^y^p  +  cr  -  p  =  a  -  4/. 


Observing  that  the  field  equation  (96)  may  be  written 
in  the  form 


R -  kg^T) 

we  get  from  (96)  the  equations, 


118  THE  MEANING  OF  RELATIVITY 

From  this  follows 


If  the  universe  is  quasi-Euclidean,  and  its  radius  of 
curvature  therefore  infinite,  then  a  would  vanish.  But 
it  is  improbable  that  the  mean  density  of  matter  in  the 
universe  is  actually  zero ;  this  is  our  third  argument 
against  the  assumption  that  the  universe  is  quasi- 
Euclidean.  Nor  does  it  seem  possible  that  our  hypo¬ 
thetical  pressure  can  vanish  ;  the  physical  nature  of  this 
pressure  can  be  appreciated  only  after  we  have  a  better 
theoretical  knowledge  of  the  electromagnetic  field. 
According  to  the  second  of  equations  (123)  the  radius, 
a,  of  the  universe  is  determined  in  terms  of  the  total 
mass,  M,  of  matter,  by  the  equation 


Mk 
47 r2 


(124) 


The  complete  dependence  of  the  geometrical  upon  the 
physical  properties  becomes  clearly  apparent  by  means 
of  this  equation. 

Thus  we  may  present  the  following  arguments  against 
the  conception  of  a  space-infinite,  and  for  the  conception 
of  a  space-bounded,  universe  : — 

I.  From  the  standpoint  of  the  theory  of  relativity, 
the  condition  for  a  closed  surface  is  very  much  simpler 
than  the  corresponding  boundary  condition  at  infinity 
of  the  quasi-Euclidean  structure  of  the  universe. 


THE  GENERAL  THEORY  119 


2.  The  idea  that  Mach  expressed,  that  inertia  depends 
upon  the  mutual  action  of  bodies,  is  contained,  to  a 
first  approximation,  in  the  equations  of  the  theory  of 
relativity ;  it  follows  from  these  equations  that  inertia 
depends,  at  least  in  part,  upon  mutual  actions  between 
masses.  As  it  is  an  unsatisfactory  assumption  to  make 
that  inertia  depends  in  part  upon  mutual  actions,  and 
in  part  upon  an  independent  property  of  space,  Mach’s 
idea  gains  in  probability.  But  this  idea  of  Mach’s 
corresponds  only  to  a  finite  universe,  bounded  in  space, 
and  not  to  a  quasi-Euclidean,  infinite  universe.  From 
the  standpoint  of  epistemology  it  is  more  satisfying  to 
have  the  mechanical  properties  of  space  completely  de¬ 
termined  by  matter,  and  this  is  the  case  only  in  a  space- 
bounded  universe. 

3.  An  infinite  universe  is  possible  only  if  the  mean 
density  of  matter  in  the  universe  vanishes.  Although 
such  an  assumption  is  logically  possible,  it  is  less  prob¬ 
able  than  the  assumption  that  there  is  a  finite  mean 
density  of  matter  in  the  universe. 


INDEX 


A 

Accelerated  masses — inductive  ac¬ 
tion  of,  1 13. 

Addition  and  subtraction  of  tensors, 
*4- 

—  theorem  of  velocities,  40. 

B 

Biot-Savart  force,  46. 

C 

Centrifugal  force,  67. 

Clocks — moving,  39. 

Compressible  viscous  fluid,  22. 
Concept  of  space,  3. 

- time,  30. 

Conditions  of  orthogonality,  7. 
Congruence — theorems  of,  3. 
Conservation  principles,  55. 
Continuum — four-dimensional,  33. 
Contraction  of  tensors,  15. 
Contra-variant  vectors,  72, 

- tensors,  75. 

Co-ordinates — preferred  systems  of, 

8. 

Co-variance  of  equation  of  contin¬ 
uity,  22. 

Co-variant,  12  et  seq. 

- vector,  72. 

Criticism  of  principle  of  inertia,  65. 
Criticisms  of  theory  of  relativity,  31. 
Curvilinear  co-ordinates,  t8. 

D 

Differentiation  of  tensors,  76,  79. 
Displacement  of  spectral  lines,  101. 

8* 


E 

Energy  and  mass,  48,  51. 

—  tensor  oi'  electromagnetic  field, 

52. 

—  —  of  matter,  56. 

Equation  of  continuity — co-variance 
of,  22. 

Equations  of  motion  of  materia 
particle,  52. 

Equivalence  of  mass  and  energy,  51. 
Equivalent  spaces  of  reference,  26. 
Euclidean  geometry,  4. 

F 

Finiteness  of  universe,  no. 

Fizeau,  29. 

Four-dimensional  continuum,  33. 
Four-vector,  43. 

Fundamental  tensor,  74. 

G 

Galilean  regions,  65. 

—  transformation,  28. 

Gauss,  68. 

Geodetic  lines,  86. 

Geometry,  Euclidean,  4. 
Gravitational  mass,  63. 

Gravitation  constant,  98. 

H 

Homogeneity  of  space,  17. 
Hydrodynamical  equations,  56. 
Hypotheses  of  pre-relativity  physics, 
77. 


121 


122  THE  MEANING  OF  RELATIVITY 


Inductive  action  of  accelerated 
masses,  113. 

Inert  and  gravitational  mass — equal¬ 
ity  of,  63. 

Invariant,  10  et  seq. 

Isotropy  of  space,  17. 

K 

Kaluza,  108. 

L 

Levi-Civita,  77. 

Light-cone,  42. 

Light  ray — path  of,  102. 

Light-time,  34. 

Linear  orthogonal  transformation,  7. 
Lorentz  electromotive  force,  46. 

—  transformation,  32. 

M 

Mach,  62,  log,  no,  m,  113,  119. 
Mass  and  Energy,  48,  51. 

—  equality  of  gravitational  and 

inert,  63. 

—  gravitational,  63. 

Maxwell’s  equations,  23. 

Mercury — perihelion  of,  103,  107. 
Michelson  and  Morley,  29. 
Minkowski,  34, 

Motion  of  particle — equations  of,  52. 
Moving  measuring  rods  and  clocks, 
39- 

Multiplication  ol  tensors,  14. 

N 

Newtonian  gravitation  constant,  98. 
O 

Operations  on  tensors,  14  et  seq. 
Orthogonality — conditions  of,  7. 
Orthogonal  transformations — linear, 
/• 


P 

Path  of  light  ray,  102. 

Perihelion  of  Mercury,  103,  107. 
Poisson’s  equation,  90. 

Preferred  systems  of  co-ordinates,  8. 
Pre-relativity  physics — hypotheses 
of,  27. 

Principle  of  equivalence,  64. 

- inertia — criticism  of,  65. 

Principles  of  conservation,  55. 

R 

Radius  of  Universe,  118. 

Rank  of  tensor,  14. 

Ray  of  light — path  of,  102. 

Reference — space  of,  4. 

Riemann,  72. 

—  tensor,  82,  85,  no. 

Rods  (measuring)  and  clocks  in  mo¬ 
tion,  39. 

Rotation,  66. 

S 

Simultaneity,  17,  30. 

Sitter,  2g. 

Skew-symmetrical  tensor,  15. 

Solar  Eclipse  expedition  (1919),  103. 
Space — concept  of,  3. 

—  homogeneity  of,  17. 

—  Isotropy  of,  17. 

Spaces  of  reference,  4  ;  equivalence 
of,  26. 

Special  Lorentz  transformation,  36. 
Spectral  lines — displacement  of,  101. 
Straightest  lines,  86. 

Stress  tensor,  22. 

Symmetrical  tensor,  15. 

Systems  of  co-ordinates — preferred, 

8. 

T 

Tensor,  12  et  seq,  72  et  seq. 

—  Addition  and  subtraction  of,  14. 

—  Contraction  of,  15. 

—  Fundamental,  74. 

—  Multiplication  of,  14. 


INDEX 


123 


Tensor,  operations,  14  et  seq. 

—  Rank  of,  14. 

—  Symmetrical  and  Skew-symmet¬ 

rical,  15. 

Tensors — formation  by  differenti¬ 
ation,  76. 

Theorem  for  addition  of  velocities, 
40. 

Theorems  of  congruence,  3. 

Theory  of  relativity,  criticisms  of,  31. 
Thirring,  113. 

Time-concept,  30. 

Time-space  concept,  33. 
Transformation — Galilean,  28. 

—  Linear  orthogonal,  7. 


U 

Universe — finiteness  of,  no. 

—  radius  of,  118. 

V 

Vector — co-variant,  72. 

—  contra-variant,  72. 

Velocities — addition  theorem  of,  40. 
Viscous  compressible  fluid,  22. 

W 

Weyl,  77,  103,  108. 


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